Angular momentum - Wikipedia

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mechanics subsection </button> <ul id="toc-Definition_in_classical_mechanics-sublist" class="vector-toc-list"> <li id="toc-Orbital_angular_momentum_in_two_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbital_angular_momentum_in_two_dimensions"> <div class="vector-toc-text"> 2.1 Orbital angular momentum in two dimensions </div> </a> <ul id="toc-Orbital_angular_momentum_in_two_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scalar_angular_momentum_from_Lagrangian_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scalar_angular_momentum_from_Lagrangian_mechanics"> <div class="vector-toc-text"> 2.2 Scalar angular momentum from Lagrangian mechanics </div> </a> <ul id="toc-Scalar_angular_momentum_from_Lagrangian_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbital_angular_momentum_in_three_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbital_angular_momentum_in_three_dimensions"> <div class="vector-toc-text"> 2.3 Orbital angular momentum in three dimensions </div> </a> <ul id="toc-Orbital_angular_momentum_in_three_dimensions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Analogy_to_linear_momentum" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Analogy_to_linear_momentum"> <div class="vector-toc-text"> 3 Analogy to linear momentum </div> </a> <button aria-controls="toc-Analogy_to_linear_momentum-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Analogy to linear momentum subsection </button> <ul id="toc-Analogy_to_linear_momentum-sublist" class="vector-toc-list"> <li id="toc-Angular_momentum_and_torque" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Angular_momentum_and_torque"> <div class="vector-toc-text"> 3.1 Angular momentum and torque </div> </a> <ul id="toc-Angular_momentum_and_torque-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Conservation_of_angular_momentum" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Conservation_of_angular_momentum"> <div class="vector-toc-text"> 4 Conservation of angular momentum </div> </a> <button aria-controls="toc-Conservation_of_angular_momentum-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Conservation of angular momentum subsection </button> <ul id="toc-Conservation_of_angular_momentum-sublist" class="vector-toc-list"> <li id="toc-General_considerations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_considerations"> <div class="vector-toc-text"> 4.1 General considerations </div> </a> <ul id="toc-General_considerations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_Newton's_second_law_of_motion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_Newton's_second_law_of_motion"> <div class="vector-toc-text"> 4.2 Relation to Newton's second law of motion </div> </a> <ul id="toc-Relation_to_Newton's_second_law_of_motion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stationary-action_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stationary-action_principle"> <div class="vector-toc-text"> 4.3 Stationary-action principle </div> </a> <ul id="toc-Stationary-action_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lagrangian_formalism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lagrangian_formalism"> <div class="vector-toc-text"> 4.4 Lagrangian formalism </div> </a> <ul id="toc-Lagrangian_formalism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hamiltonian_formalism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hamiltonian_formalism"> <div class="vector-toc-text"> 4.5 Hamiltonian formalism </div> </a> <ul id="toc-Hamiltonian_formalism-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Angular_momentum_in_orbital_mechanics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Angular_momentum_in_orbital_mechanics"> <div class="vector-toc-text"> 5 Angular momentum in orbital mechanics </div> </a> <ul id="toc-Angular_momentum_in_orbital_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solid_bodies" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Solid_bodies"> <div class="vector-toc-text"> 6 Solid bodies </div> </a> <button aria-controls="toc-Solid_bodies-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Solid bodies subsection </button> <ul id="toc-Solid_bodies-sublist" class="vector-toc-list"> <li id="toc-Collection_of_particles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Collection_of_particles"> <div class="vector-toc-text"> 6.1 Collection of particles </div> </a> <ul id="toc-Collection_of_particles-sublist" class="vector-toc-list"> <li id="toc-Single_particle_case" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Single_particle_case"> <div class="vector-toc-text"> 6.1.1 Single particle case </div> </a> <ul id="toc-Single_particle_case-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Case_of_a_fixed_center_of_mass" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Case_of_a_fixed_center_of_mass"> <div class="vector-toc-text"> 6.1.2 Case of a fixed center of mass </div> </a> <ul id="toc-Case_of_a_fixed_center_of_mass-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Angular_momentum_in_general_relativity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Angular_momentum_in_general_relativity"> <div class="vector-toc-text"> 7 Angular momentum in general relativity </div> </a> <ul id="toc-Angular_momentum_in_general_relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angular_momentum_in_quantum_mechanics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Angular_momentum_in_quantum_mechanics"> <div class="vector-toc-text"> 8 Angular momentum in quantum mechanics </div> </a> <button aria-controls="toc-Angular_momentum_in_quantum_mechanics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Angular momentum in quantum mechanics subsection </button> <ul id="toc-Angular_momentum_in_quantum_mechanics-sublist" class="vector-toc-list"> <li id="toc-Spin,_orbital,_and_total_angular_momentum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spin,_orbital,_and_total_angular_momentum"> <div class="vector-toc-text"> 8.1 Spin, orbital, and total angular momentum </div> </a> <ul id="toc-Spin,_orbital,_and_total_angular_momentum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantization"> <div class="vector-toc-text"> 8.2 Quantization </div> </a> <ul id="toc-Quantization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uncertainty" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uncertainty"> <div class="vector-toc-text"> 8.3 Uncertainty </div> </a> <ul id="toc-Uncertainty-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Total_angular_momentum_as_generator_of_rotations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Total_angular_momentum_as_generator_of_rotations"> <div class="vector-toc-text"> 8.4 Total angular momentum as generator of rotations </div> </a> <ul id="toc-Total_angular_momentum_as_generator_of_rotations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Angular_momentum_in_electrodynamics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Angular_momentum_in_electrodynamics"> <div class="vector-toc-text"> 9 Angular momentum in electrodynamics </div> </a> <ul id="toc-Angular_momentum_in_electrodynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angular_momentum_in_optics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Angular_momentum_in_optics"> <div class="vector-toc-text"> 10 Angular momentum in optics </div> </a> <ul id="toc-Angular_momentum_in_optics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angular_momentum_in_nature_and_the_cosmos" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Angular_momentum_in_nature_and_the_cosmos"> <div class="vector-toc-text"> 11 Angular momentum in nature and the cosmos </div> </a> <ul id="toc-Angular_momentum_in_nature_and_the_cosmos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angular_momentum_in_engineering_and_technology" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Angular_momentum_in_engineering_and_technology"> <div class="vector-toc-text"> 12 Angular momentum in engineering and technology </div> </a> <ul id="toc-Angular_momentum_in_engineering_and_technology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 13 History </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle History subsection </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-The_Law_of_Areas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Law_of_Areas"> <div class="vector-toc-text"> 13.1 The Law of Areas </div> </a> <ul id="toc-The_Law_of_Areas-sublist" class="vector-toc-list"> <li id="toc-Newton's_derivation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Newton's_derivation"> <div class="vector-toc-text"> 13.1.1 Newton's derivation </div> </a> <ul id="toc-Newton's_derivation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conservation_of_angular_momentum_in_the_Law_of_Areas" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conservation_of_angular_momentum_in_the_Law_of_Areas"> <div class="vector-toc-text"> 13.1.2 Conservation of angular momentum in the Law of Areas </div> </a> <ul id="toc-Conservation_of_angular_momentum_in_the_Law_of_Areas-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-After_Newton" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#After_Newton"> <div class="vector-toc-text"> 13.2 After Newton </div> </a> <ul id="toc-After_Newton-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 14 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 15 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 16 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 17 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Angular momentum</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 73 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-73" aria-hidden="true" > 73 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Hoekmomentum" title="Hoekmomentum – Afrikaans" lang="af" hreflang="af" data-title="Hoekmomentum" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D8%AE%D9%85_%D8%B2%D8%A7%D9%88%D9%8A" title="زخم زاوي – Arabic" lang="ar" hreflang="ar" data-title="زخم زاوي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Momentu_angular" title="Momentu angular – Asturian" lang="ast" hreflang="ast" data-title="Momentu angular" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0mpuls_momenti" title="İmpuls momenti – Azerbaijani" lang="az" hreflang="az" data-title="İmpuls momenti" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8C%E0%A6%A3%E0%A6%BF%E0%A6%95_%E0%A6%AD%E0%A6%B0%E0%A6%AC%E0%A7%87%E0%A6%97" title="কৌণিক ভরবেগ – Bangla" lang="bn" hreflang="bn" data-title="কৌণিক ভরবেগ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B0%D0%BD%D1%82_%D1%96%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81%D1%83" title="Момант імпульсу – Belarusian" lang="be" hreflang="be" data-title="Момант імпульсу" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%81%D0%B0" title="Момент на импулса – Bulgarian" lang="bg" hreflang="bg" data-title="Момент на импулса" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Ugaona_koli%C4%8Dina_kretanja" title="Ugaona količina kretanja – Bosnian" lang="bs" hreflang="bs" data-title="Ugaona količina kretanja" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Moment_angular" title="Moment angular – Catalan" lang="ca" hreflang="ca" data-title="Moment angular" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81_%D1%81%D0%B0%D0%BC%D0%B0%D0%BD%D1%87%C4%95" title="Импульс саманчĕ – Chuvash" lang="cv" hreflang="cv" data-title="Импульс саманчĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Moment_hybnosti" title="Moment hybnosti – Czech" lang="cs" hreflang="cs" data-title="Moment hybnosti" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Impulsmoment" title="Impulsmoment – Danish" lang="da" hreflang="da" data-title="Impulsmoment" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Drehimpuls" title="Drehimpuls – German" lang="de" hreflang="de" data-title="Drehimpuls" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Impulsimoment" title="Impulsimoment – Estonian" lang="et" hreflang="et" data-title="Impulsimoment" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%84%CF%81%CE%BF%CF%86%CE%BF%CF%81%CE%BC%CE%AE" title="Στροφορμή – Greek" lang="el" hreflang="el" data-title="Στροφορμή" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Momento_angular" title="Momento angular – Spanish" lang="es" hreflang="es" data-title="Momento angular" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Angula_movokvanto" title="Angula movokvanto – Esperanto" lang="eo" hreflang="eo" data-title="Angula movokvanto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Momentu_angeluar" title="Momentu angeluar – Basque" lang="eu" hreflang="eu" data-title="Momentu angeluar" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%DA%A9%D8%A7%D9%86%D9%87_%D8%B2%D8%A7%D9%88%DB%8C%D9%87%E2%80%8C%D8%A7%DB%8C" title="تکانه زاویه‌ای – Persian" lang="fa" hreflang="fa" data-title="تکانه زاویه‌ای" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Moment_cin%C3%A9tique" title="Moment cinétique – French" lang="fr" hreflang="fr" data-title="Moment cinétique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/M%C3%B3iminteam_uilleach" title="Móiminteam uilleach – Irish" lang="ga" hreflang="ga" data-title="Móiminteam uilleach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Momento_angular" title="Momento angular – Galician" lang="gl" hreflang="gl" data-title="Momento angular" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B0%81%EC%9A%B4%EB%8F%99%EB%9F%89" title="각운동량 – Korean" lang="ko" hreflang="ko" data-title="각운동량" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D5%B4%D5%BA%D5%B8%D6%82%D5%AC%D5%BD%D5%AB_%D5%B4%D5%B8%D5%B4%D5%A5%D5%B6%D5%BF" title="Իմպուլսի մոմենտ – Armenian" lang="hy" hreflang="hy" data-title="Իմպուլսի մոմենտ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%8B%E0%A4%A3%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%B5%E0%A5%87%E0%A4%97" title="कोणीय संवेग – Hindi" lang="hi" hreflang="hi" data-title="कोणीय संवेग" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kutna_koli%C4%8Dina_gibanja" title="Kutna količina gibanja – Croatian" lang="hr" hreflang="hr" data-title="Kutna količina gibanja" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Momentum_sudut" title="Momentum sudut – Indonesian" lang="id" hreflang="id" data-title="Momentum sudut" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hverfi%C3%BEungi" title="Hverfiþungi – Icelandic" lang="is" hreflang="is" data-title="Hverfiþungi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Momento_angolare" title="Momento angolare – Italian" lang="it" hreflang="it" data-title="Momento angolare" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%A0%D7%A2_%D7%96%D7%95%D7%95%D7%99%D7%AA%D7%99" title="תנע זוויתי – Hebrew" lang="he" hreflang="he" data-title="תנע זוויתי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%98%E1%83%9B%E1%83%9E%E1%83%A3%E1%83%9A%E1%83%A1%E1%83%98%E1%83%A1_%E1%83%9B%E1%83%9D%E1%83%9B%E1%83%94%E1%83%9C%E1%83%A2%E1%83%98" title="იმპულსის მომენტი – Georgian" lang="ka" hreflang="ka" data-title="იმპულსის მომენტი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D1%96" title="Импульс моменті – Kazakh" lang="kk" hreflang="kk" data-title="Импульс моменті" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Impulsa_moments" title="Impulsa moments – Latvian" lang="lv" hreflang="lv" data-title="Impulsa moments" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Judesio_kiekio_momentas" title="Judesio kiekio momentas – Lithuanian" lang="lt" hreflang="lt" data-title="Judesio kiekio momentas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Perd%C3%BClet" title="Perdület – Hungarian" lang="hu" hreflang="hu" data-title="Perdület" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%81%D0%BE%D1%82" title="Момент на импулсот – Macedonian" lang="mk" hreflang="mk" data-title="Момент на импулсот" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A5%8B%E0%A4%A8%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%B5%E0%A5%87%E0%A4%97" title="कोनीय संवेग – Marathi" lang="mr" hreflang="mr" data-title="कोनीय संवेग" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Momentum_sudut" title="Momentum sudut – Malay" lang="ms" hreflang="ms" data-title="Momentum sudut" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%91%E1%80%B1%E1%80%AC%E1%80%84%E1%80%B7%E1%80%BA%E1%80%95%E1%80%BC%E1%80%B1%E1%80%AC%E1%80%84%E1%80%BA%E1%80%B8%E1%80%A1%E1%80%9F%E1%80%AF%E1%80%94%E1%80%BA" title="ထောင့်ပြောင်းအဟုန် – Burmese" lang="my" hreflang="my" data-title="ထောင့်ပြောင်းအဟုန်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Impulsmoment" title="Impulsmoment – Dutch" lang="nl" hreflang="nl" data-title="Impulsmoment" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A7%92%E9%81%8B%E5%8B%95%E9%87%8F" title="角運動量 – Japanese" lang="ja" hreflang="ja" data-title="角運動量" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Dreiimpuls" title="Dreiimpuls – Northern Frisian" lang="frr" hreflang="frr" data-title="Dreiimpuls" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Drivmoment" title="Drivmoment – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Drivmoment" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Vinkelmoment" title="Vinkelmoment – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Vinkelmoment" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Moment_cinetic" title="Moment cinetic – Occitan" lang="oc" hreflang="oc" data-title="Moment cinetic" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%90%E0%A8%82%E0%A8%97%E0%A9%81%E0%A8%B2%E0%A8%B0_%E0%A8%AE%E0%A9%8B%E0%A8%AE%E0%A9%88%E0%A8%82%E0%A8%9F%E0%A8%AE" title="ਐਂਗੁਲਰ ਮੋਮੈਂਟਮ – Punjabi" lang="pa" hreflang="pa" data-title="ਐਂਗੁਲਰ ਮੋਮੈਂਟਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Moment_angolar" title="Moment angolar – Piedmontese" lang="pms" hreflang="pms" data-title="Moment angolar" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Moment_p%C4%99du" title="Moment pędu – Polish" lang="pl" hreflang="pl" data-title="Moment pędu" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Momento_angular" title="Momento angular – Portuguese" lang="pt" hreflang="pt" data-title="Momento angular" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Moment_cinetic" title="Moment cinetic – Romanian" lang="ro" hreflang="ro" data-title="Moment cinetic" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81%D0%B0" title="Момент импульса – Russian" lang="ru" hreflang="ru" data-title="Момент импульса" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Impulsi_k%C3%ABndor" title="Impulsi këndor – Albanian" lang="sq" hreflang="sq" data-title="Impulsi këndor" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Angular_momentum" title="Angular momentum – Simple English" lang="en-simple" hreflang="en-simple" data-title="Angular momentum" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Moment_hybnosti" title="Moment hybnosti – Slovak" lang="sk" hreflang="sk" data-title="Moment hybnosti" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Vrtilna_koli%C4%8Dina" title="Vrtilna količina – Slovenian" lang="sl" hreflang="sl" data-title="Vrtilna količina" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%81%D0%B0" title="Момент импулса – Serbian" lang="sr" hreflang="sr" data-title="Момент импулса" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Ugaoni_moment" title="Ugaoni moment – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Ugaoni moment" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Mom%C3%A9ntum_sudut" title="Moméntum sudut – Sundanese" lang="su" hreflang="su" data-title="Moméntum sudut" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Py%C3%B6rimism%C3%A4%C3%A4r%C3%A4" title="Pyörimismäärä – Finnish" lang="fi" hreflang="fi" data-title="Pyörimismäärä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/R%C3%B6relsem%C3%A4ngdsmoment" title="Rörelsemängdsmoment – Swedish" lang="sv" hreflang="sv" data-title="Rörelsemängdsmoment" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%B3%E0%AF%88%E0%AE%B5%E0%AF%81%E0%AE%A8%E0%AF%8D%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="வளைவுந்தம் – Tamil" lang="ta" hreflang="ta" data-title="வளைவுந்தம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Imir_u%C9%A3mir" title="Imir uɣmir – Kabyle" lang="kab" hreflang="kab" data-title="Imir uɣmir" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%C4%B0mpuls_moment%C4%B1" title="İmpuls momentı – Tatar" lang="tt" hreflang="tt" data-title="İmpuls momentı" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%95%E0%B1%8B%E0%B0%A3%E0%B1%80%E0%B0%AF_%E0%B0%A6%E0%B1%8D%E0%B0%B0%E0%B0%B5%E0%B1%8D%E0%B0%AF%E0%B0%B5%E0%B1%87%E0%B0%97%E0%B0%82" title="కోణీయ ద్రవ్యవేగం – Telugu" lang="te" hreflang="te" data-title="కోణీయ ద్రవ్యవేగం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%82%E0%B8%A1%E0%B9%80%E0%B8%A1%E0%B8%99%E0%B8%95%E0%B8%B1%E0%B8%A1%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%A1%E0%B8%B8%E0%B8%A1" title="โมเมนตัมเชิงมุม – Thai" lang="th" hreflang="th" data-title="โมเมนตัมเชิงมุม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/A%C3%A7%C4%B1sal_momentum" title="Açısal momentum – Turkish" lang="tr" hreflang="tr" data-title="Açısal momentum" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D1%96%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81%D1%83" title="Момент імпульсу – Ukrainian" lang="uk" hreflang="uk" data-title="Момент імпульсу" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B2%D8%A7%D9%88%DB%8C%D8%A7%D8%A6%DB%8C_%D9%85%D8%B9%DB%8C%D8%A7%D8%B1_%D8%AD%D8%B1%DA%A9%D8%AA" title="زاویائی معیار حرکت – Urdu" lang="ur" hreflang="ur" data-title="زاویائی معیار حرکت" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%C3%B4_men_%C4%91%E1%BB%99ng_l%C6%B0%E1%BB%A3ng" title="Mô men động lượng – Vietnamese" lang="vi" hreflang="vi" data-title="Mô men động lượng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%A7%92%E5%8B%95%E9%87%8F" title="角動量 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="角動量" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%A7%92%E5%8A%A8%E9%87%8F" title="角动量 – Wu" lang="wuu" hreflang="wuu" data-title="角动量" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%A7%92%E5%8B%95%E9%87%8F" title="角動量 – Cantonese" lang="yue" hreflang="yue" data-title="角動量" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%A7%92%E5%8A%A8%E9%87%8F" title="角动量 – Chinese" lang="zh" hreflang="zh" data-title="角动量" 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Conserved physical quantity; rotational analogue of linear momentum</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Angular momentum</th></tr><tr><td colspan="2" class="infobox-image"><a href="/wiki/File:Gyroskop.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Gyroskop.jpg/220px-Gyroskop.jpg" decoding="async" width="220" height="346" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/b9/Gyroskop.jpg 1.5x" data-file-width="282" data-file-height="444" /></a><div class="infobox-caption">This <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a> remains upright while spinning owing to the conservation of its angular momentum.</div></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data">L</td></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/SI_base_unit" title="SI base unit">SI base units</a></th><td class="infobox-data">kg⋅m2⋅s−1</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Conserved_quantity" title="Conserved quantity">Conserved</a>?</th><td class="infobox-data">yes</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from other quantities</div></th><td class="infobox-data">L = Iω = r × p</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0484fb7c4e4306a940c19093b216b3a88974f49f" 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-left:0.9em;padding-right:0.9em;"><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></th></tr><tr><td class="sidebar-image"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">F</mtext> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ad0a6d6780c3abc5247abd82bd8a2249d56ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.318ex; height:5.509ex;" alt="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"><div class="sidebar-caption" style="font-size:90%;padding:0.6em 0;font-style:italic;"><a href="/wiki/Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</a></div></td></tr><tr><th class="sidebar-heading" style="font-weight: bold; display:block;margin-bottom:1.0em;"> <div class="hlist"> <ul><li><a href="/wiki/History_of_classical_mechanics" title="History of classical mechanics">History</a></li> <li><a href="/wiki/Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline</a></li> <li><a href="/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">Textbooks</a></li></ul> </div></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Applied_mechanics" title="Applied mechanics">Applied</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li> <li><a href="/wiki/Analytical_dynamics" class="mw-redirect" title="Analytical dynamics">Dynamics</a></li> <li><a href="/wiki/Classical_field_theory" title="Classical field theory">Field theory</a></li> <li><a href="/wiki/Kinematics" title="Kinematics">Kinematics</a></li> <li><a href="/wiki/Kinetics_(physics)" title="Kinetics (physics)">Kinetics</a></li> <li><a href="/wiki/Statics" title="Statics">Statics</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Acceleration" title="Acceleration">Acceleration</a></li> <li><a class="mw-selflink selflink">Angular momentum</a></li> <li><a href="/wiki/Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li> <li><a href="/wiki/D%27Alembert%27s_principle" title="D'Alembert's principle">D'Alembert's principle</a></li> <li><a href="/wiki/Energy" title="Energy">Energy</a> <ul><li><a href="/wiki/Kinetic_energy#Newtonian_kinetic_energy" title="Kinetic energy">kinetic</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">potential</a></li></ul></li> <li><a href="/wiki/Force" title="Force">Force</a></li> <li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li> <li><a href="/wiki/Impulse_(physics)" title="Impulse (physics)">Impulse</a></li> <li><a href="/wiki/Inertia" title="Inertia">Inertia</a> / <a href="/wiki/Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></li> <li><a href="/wiki/Mass" title="Mass">Mass</a></li> <li> <a href="/wiki/Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Mechanical work</a></li> <li> <a href="/wiki/Moment_(physics)" title="Moment (physics)">Moment</a></li> <li><a href="/wiki/Momentum" title="Momentum">Momentum</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/Speed" title="Speed">Speed</a></li> <li><a href="/wiki/Time" title="Time">Time</a></li> <li><a href="/wiki/Torque" title="Torque">Torque</a></li> <li><a href="/wiki/Velocity" title="Velocity">Velocity</a></li> <li><a href="/wiki/Virtual_work" title="Virtual work">Virtual work</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"> <ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></div></li> <li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a> <div class="plainlist"><ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="/wiki/Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="/wiki/Appell%27s_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Core topics</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Damping" title="Damping">Damping</a></li> <li><a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li> <li><a href="/wiki/Equations_of_motion" title="Equations of motion">Equations of motion</a></li> <li><a href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion">Euler's laws of motion</a></li> <li><a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a></li> <li><a href="/wiki/Friction" title="Friction">Friction</a></li> <li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul> </div> <ul><li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a> / <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Motion" title="Motion">Motion</a> (<a href="/wiki/Linear_motion" title="Linear motion">linear</a>)</li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li> <li><a href="/wiki/Relative_velocity" title="Relative velocity">Relative velocity</a></li> <li><a href="/wiki/Rigid_body" title="Rigid body">Rigid body</a> <ul><li><a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li> <li><a href="/wiki/Euler%27s_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li> <li><a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li> <li><a href="/wiki/Vibration" title="Vibration">Vibration</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)"><a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Circular_motion" title="Circular motion">Circular motion</a></li> <li><a href="/wiki/Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li> <li><a href="/wiki/Centripetal_force" title="Centripetal force">Centripetal force</a></li> <li><a href="/wiki/Centrifugal_force" title="Centrifugal force">Centrifugal force</a> <ul><li><a href="/wiki/Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li> <li><a href="/wiki/Tangential_speed" title="Tangential speed">Tangential speed</a></li> <li><a href="/wiki/Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul> </div> <ul><li><a href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="/wiki/Angular_displacement" title="Angular displacement">displacement</a> / <a href="/wiki/Angular_frequency" title="Angular frequency">frequency</a> / <a href="/wiki/Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a></li> <li><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li> <li><a href="/wiki/Edmond_Halley" title="Edmond Halley">Halley</a></li> <li><a href="/wiki/Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li> <li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li> <li><a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">d'Alembert</a></li> <li><a href="/wiki/Alexis_Clairaut" title="Alexis Clairaut">Clairaut</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li> <li><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a></li> <li><a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Edward_Routh" title="Edward Routh">Routh</a></li> <li><a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Liouville</a></li> <li><a href="/wiki/Paul_%C3%89mile_Appell" title="Paul Émile Appell">Appell</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Bernard_Koopman" title="Bernard Koopman">Koopman</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <ul><li><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/14px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="14" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/21px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/28px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a> <a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Classical_mechanics" title="Category:Classical mechanics">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classical_mechanics" title="Template:Classical mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classical_mechanics" title="Template talk:Classical mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics" title="Special:EditPage/Template:Classical mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> Angular momentum (sometimes called moment of momentum or rotational momentum) is the <a href="/wiki/Rotational" class="mw-redirect" title="Rotational">rotational</a> analog of <a href="/wiki/Momentum" title="Momentum">linear momentum</a>. It is an important <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantity</a> because it is a <a href="/wiki/Conservation_law" title="Conservation law">conserved quantity</a> – the total angular momentum of a <a href="/wiki/Closed_system" title="Closed system">closed system</a> remains constant. Angular momentum has both a <a href="/wiki/Direction_(geometry)" title="Direction (geometry)">direction</a> and a magnitude, and both are conserved. <a href="/wiki/Bicycle_and_motorcycle_dynamics" title="Bicycle and motorcycle dynamics">Bicycles and motorcycles</a>, <a href="/wiki/Flying_disc" class="mw-redirect" title="Flying disc">flying discs</a>,<a href="#cite_note-1">[1]</a> <a href="/wiki/Rifling" title="Rifling">rifled bullets</a>, and <a href="/wiki/Gyroscope" title="Gyroscope">gyroscopes</a> owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why <a href="/wiki/Hurricane" class="mw-redirect" title="Hurricane">hurricanes</a><a href="#cite_note-Tropical_Cyclone_Structure-2">[2]</a> form spirals and <a href="/wiki/Neutron_star" title="Neutron star">neutron stars</a> have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it. The three-dimensional angular momentum for a <a href="/wiki/Point_particle" title="Point particle">point particle</a> is classically represented as a <a href="/wiki/Pseudovector" title="Pseudovector">pseudovector</a> r × p, the <a href="/wiki/Cross_product" title="Cross product">cross product</a> of the particle's <a href="/wiki/Position_vector" class="mw-redirect" title="Position vector">position vector</a> r (relative to some origin) and its <a href="/wiki/Momentum_vector" class="mw-redirect" title="Momentum vector">momentum vector</a>; the latter is p = mv in <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a>. Unlike linear momentum, angular momentum depends on where this origin is chosen, since the particle's position is measured from it. Angular momentum is an <a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">extensive quantity</a>; that is, the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuous</a> <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a> or a <a href="/wiki/Fluid" title="Fluid">fluid</a>, the total angular momentum is the <a href="/wiki/Volume_integral" title="Volume integral">volume integral</a> of angular momentum density (angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body. Similar to conservation of linear momentum, where it is conserved if there is no external force, angular momentum is conserved if there is no external <a href="/wiki/Torque" title="Torque">torque</a>. Torque can be defined as the rate of change of angular momentum, analogous to <a href="/wiki/Force" title="Force">force</a>. The net external torque on any system is always equal to the total torque on the system; the sum of all internal torques of any system is always 0 (this is the rotational analogue of <a href="/wiki/Newton%27s_third_law_of_motion" class="mw-redirect" title="Newton's third law of motion">Newton's third law of motion</a>). Therefore, for a <a href="/wiki/Closed_system" title="Closed system">closed system</a> (where there is no net external torque), the total torque on the system must be 0, which means that the total angular momentum of the system is constant. The change in angular momentum for a particular interaction is called angular impulse, sometimes twirl.<a href="#cite_note-3">[3]</a> Angular impulse is the angular analog of (linear) <a href="/wiki/Impulse_(physics)" title="Impulse (physics)">impulse</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=1" title="Edit section: Examples">edit</a>]</div> The trivial case of the angular momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> of a body in an orbit is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=2\pi Mfr^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>M</mi> <mi>f</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=2\pi Mfr^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd6fb27acbb8fd24014b59c275a896253974c4c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.999ex; height:3.009ex;" alt="{\displaystyle L=2\pi Mfr^{2}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is the <a href="/wiki/Mass" title="Mass">mass</a> of the orbiting object, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is the orbit's <a href="/wiki/Frequency" title="Frequency">frequency</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> is the orbit's radius. The angular momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> of a uniform rigid sphere rotating around its axis, instead, is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\frac {4}{5}}\pi Mfr^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mi>π</mi> <mi>M</mi> <mi>f</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {4}{5}}\pi Mfr^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d46dac3e3c9558e0278430f6173bacba1ac87882" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.836ex; height:5.176ex;" alt="{\displaystyle L={\frac {4}{5}}\pi Mfr^{2}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is the sphere's mass, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is the frequency of rotation and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> is the sphere's radius. Thus, for example, the orbital angular momentum of the <a href="/wiki/Earth" title="Earth">Earth</a> with respect to the Sun is about 2.66 × 1040 kg⋅m2⋅s−1, while its rotational angular momentum is about 7.05 × 1033 kg⋅m2⋅s−1. In the case of a uniform rigid sphere rotating around its axis, if, instead of its mass, its <a href="/wiki/Density" title="Density">density</a> is known, the angular momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\frac {16}{15}}\pi ^{2}\rho fr^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>16</mn> <mn>15</mn> </mfrac> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mi>f</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {16}{15}}\pi ^{2}\rho fr^{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb69c21fbd3b214719b1277a236069581cbedf90" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.814ex; height:5.176ex;" alt="{\displaystyle L={\frac {16}{15}}\pi ^{2}\rho fr^{5}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> is the sphere's <a href="/wiki/Density" title="Density">density</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is the frequency of rotation and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> is the sphere's radius. In the simplest case of a spinning disk, the angular momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> is given by<a href="#cite_note-hyperphysics-4">[4]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\pi Mfr^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>π</mi> <mi>M</mi> <mi>f</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\pi Mfr^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc1488fa69331cde0e30b83bde073268ed41362" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.837ex; height:3.009ex;" alt="{\displaystyle L=\pi Mfr^{2}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is the disk's mass, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is the frequency of rotation and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> is the disk's radius. If instead the disk rotates about its diameter (e.g. coin toss), its angular momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> is given by<a href="#cite_note-hyperphysics-4">[4]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\frac {1}{2}}\pi Mfr^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mi>M</mi> <mi>f</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {1}{2}}\pi Mfr^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc62b58c37cf04550d015cfa06903d6eba42d154" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.836ex; height:5.176ex;" alt="{\displaystyle L={\frac {1}{2}}\pi Mfr^{2}}"> <div class="mw-heading mw-heading2"><h2 id="Definition_in_classical_mechanics">Definition in classical mechanics</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=2" title="Edit section: Definition in classical mechanics">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Orbital angular momentum" redirects here. For other uses, see <a href="/wiki/Orbital_angular_momentum_(disambiguation)" class="mw-disambig" title="Orbital angular momentum (disambiguation)">Orbital angular momentum (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Spin angular momentum" redirects here. For other uses, see <a href="/wiki/Spin_angular_momentum_(disambiguation)" class="mw-disambig" title="Spin angular momentum (disambiguation)">Spin angular momentum (disambiguation)</a>.</div> Just as for <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a>, there are two special types of angular <a href="/wiki/Momentum" title="Momentum">momentum</a> of an object: the spin angular momentum is the angular momentum about the object's <a href="/wiki/Centre_of_mass" class="mw-redirect" title="Centre of mass">centre of mass</a>, while the orbital angular momentum is the angular momentum about a chosen center of rotation. The <a href="/wiki/Earth" title="Earth">Earth</a> has an orbital angular momentum by nature of revolving around the <a href="/wiki/Sun" title="Sun">Sun</a>, and a spin angular momentum by nature of its daily rotation around the polar axis. The total angular momentum is the sum of the spin and orbital angular momenta. In the case of the Earth the primary conserved quantity is the total angular momentum of the solar system because angular momentum is exchanged to a small but important extent among the planets and the Sun. The orbital angular momentum vector of a point particle is always parallel and directly proportional to its orbital angular <a href="/wiki/Velocity" title="Velocity">velocity</a> vector ω, where the constant of proportionality depends on both the mass of the particle and its distance from origin. The spin angular momentum vector of a rigid body is proportional but not always parallel to the spin angular velocity vector Ω, making the constant of proportionality a second-rank <a href="/wiki/Tensor" title="Tensor">tensor</a> rather than a scalar. <div class="mw-heading mw-heading3"><h3 id="Orbital_angular_momentum_in_two_dimensions">Orbital angular momentum in two dimensions</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=3" title="Edit section: Orbital angular momentum in two dimensions">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ang_mom_2d.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Ang_mom_2d.png/220px-Ang_mom_2d.png" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Ang_mom_2d.png/330px-Ang_mom_2d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Ang_mom_2d.png/440px-Ang_mom_2d.png 2x" data-file-width="698" data-file-height="682" /></a><figcaption><a href="/wiki/Velocity" title="Velocity">Velocity</a> of the <a href="/wiki/Particle" title="Particle">particle</a> m with respect to the origin O can be resolved into components parallel to (v∥) and perpendicular to (v⊥) the radius vector r. The angular momentum of m is proportional to the <a href="/wiki/Perpendicular_component" class="mw-redirect" title="Perpendicular component">perpendicular component</a> v⊥ of the velocity, or equivalently, to the perpendicular distance r⊥ from the origin.</figcaption></figure> Angular momentum is a <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> quantity (more precisely, a <a href="/wiki/Pseudovector" title="Pseudovector">pseudovector</a>) that represents the product of a body's <a href="/wiki/Moment_of_inertia" title="Moment of inertia">rotational inertia</a> and <a href="/wiki/Angular_velocity" title="Angular velocity">rotational velocity</a> (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>, it is sufficient to discard the vector nature of angular momentum, and treat it as a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> (more precisely, a <a href="/wiki/Pseudoscalar" title="Pseudoscalar">pseudoscalar</a>).<a href="#cite_note-Wilson-5">[5]</a> Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum p is proportional to <a href="/wiki/Mass" title="Mass">mass</a> m and <a href="/wiki/Speed" title="Speed">linear speed</a> v, <a href="/w/index.php?title=Special:MathWikibase&qid=Q41273" style="color:inherit;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=mv,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>m</mi> <mi>v</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=mv,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4b93eb594373b6fa496b126057996eefce3364" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.172ex; height:2.009ex;" alt="{\displaystyle p=mv,}"></a> angular momentum L is proportional to <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> I and <a href="/wiki/Angular_frequency" title="Angular frequency">angular speed</a> ω measured in radians per second.<a href="#cite_note-Worthington-6">[6]</a> <a href="/w/index.php?title=Special:MathWikibase&qid=Q161254" style="color:inherit;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=I\omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>I</mi> <mi>ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=I\omega .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ce1597dcef54047a562d9108835bcbd8358d32" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.946ex; height:2.176ex;" alt="{\displaystyle L=I\omega .}"></a> Unlike mass, which depends only on amount of matter, moment of inertia depends also on the position of the axis of rotation and the distribution of the matter. Unlike linear velocity, which does not depend upon the choice of origin, orbital angular velocity is always measured with respect to a fixed origin. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center.<a href="#cite_note-Taylor90-7">[7]</a> In the case of circular motion of a single particle, we can use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=r^{2}m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=r^{2}m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce8ff961fab0cdfe7ac50e8bee6d6ef1ee62347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;" alt="{\displaystyle I=r^{2}m}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={v}/{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={v}/{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d74ee5180d273d318e51935cdcf89fc155de092" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.883ex; height:2.843ex;" alt="{\displaystyle \omega ={v}/{r}}"> to expand angular momentum as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=r^{2}m\cdot {v}/{r},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=r^{2}m\cdot {v}/{r},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a5d9ca58b6c02f3910eb59baaa6467d2b40fe45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.489ex; height:3.176ex;" alt="{\displaystyle L=r^{2}m\cdot {v}/{r},}"> reducing to: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=rmv,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>r</mi> <mi>m</mi> <mi>v</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=rmv,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce9be937fd731eccda69830a7cf2128deecb0ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.545ex; height:2.509ex;" alt="{\displaystyle L=rmv,}"> the product of the <a href="/wiki/Radius" title="Radius">radius</a> of rotation r and the linear momentum of the particle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=mv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>m</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=mv}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2acbe7154884d4dbe30b9a0b399e43cefd8654c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.525ex; height:2.009ex;" alt="{\displaystyle p=mv}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=r\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>r</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=r\omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23b5b5c066f5be8dee84c4cb33dea2383de8b012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.721ex; height:1.676ex;" alt="{\displaystyle v=r\omega }"> is the <a href="/wiki/Speed#Tangential_speed" title="Speed">linear (tangential) speed</a>. This simple analysis can also apply to non-circular motion if one uses the component of the motion <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to the <a href="/wiki/Radius_vector" class="mw-redirect" title="Radius vector">radius vector</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=rmv_{\perp },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>r</mi> <mi>m</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=rmv_{\perp },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54cba40e6e82b32083735501d13f41e68cccf06d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.056ex; height:2.509ex;" alt="{\displaystyle L=rmv_{\perp },}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\perp }=v\sin(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> <mo>=</mo> <mi>v</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\perp }=v\sin(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163f5f644793a0f6eb62286bb838848317b0a8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.007ex; height:2.843ex;" alt="{\displaystyle v_{\perp }=v\sin(\theta )}"> is the perpendicular component of the motion. Expanding, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=rmv\sin(\theta ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>r</mi> <mi>m</mi> <mi>v</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=rmv\sin(\theta ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f093a6fdb253e9d684af3f225239b92697cba364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.687ex; height:2.843ex;" alt="{\displaystyle L=rmv\sin(\theta ),}"> rearranging, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=r\sin(\theta )mv,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>m</mi> <mi>v</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=r\sin(\theta )mv,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e238548108c2e9e0c3ea696da8500c9885015c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.687ex; height:2.843ex;" alt="{\displaystyle L=r\sin(\theta )mv,}"> and reducing, angular momentum can also be expressed, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=r_{\perp }mv,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> <mi>m</mi> <mi>v</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=r_{\perp }mv,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb8528a226fa2915919bed41269ad720065d1352" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.056ex; height:2.509ex;" alt="{\displaystyle L=r_{\perp }mv,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\perp }=r\sin(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\perp }=r\sin(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e429dae54c4064967213f2be1cf6170f814e664" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.849ex; height:2.843ex;" alt="{\displaystyle r_{\perp }=r\sin(\theta )}"> is the length of the <a href="/wiki/Torque#Moment_arm_formula" title="Torque">moment arm</a>, a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, (length of moment arm) × (linear momentum), to which the term moment of momentum refers.<a href="#cite_note-Dadourian-8">[8]</a> <div class="mw-heading mw-heading3"><h3 id="Scalar_angular_momentum_from_Lagrangian_mechanics">Scalar angular momentum from Lagrangian mechanics</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=4" title="Edit section: Scalar angular momentum from Lagrangian mechanics">edit</a>]</div> Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum) of the angular coordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> expressed in the <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a> of the mechanical system. Consider a mechanical system with a mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> constrained to move in a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> in the absence of any external force field. The kinetic energy of the system is <a href="/w/index.php?title=Special:MathWikibase&qid=Q46276" style="color:inherit;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\tfrac {1}{2}}mr^{2}\omega ^{2}={\tfrac {1}{2}}mr^{2}{\dot {\phi }}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\tfrac {1}{2}}mr^{2}\omega ^{2}={\tfrac {1}{2}}mr^{2}{\dot {\phi }}^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50680c8ca5ae379b8f0f1cc1371d249004196d68" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.105ex; height:4.009ex;" alt="{\displaystyle T={\tfrac {1}{2}}mr^{2}\omega ^{2}={\tfrac {1}{2}}mr^{2}{\dot {\phi }}^{2}.}"></a> And the potential energy is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46118843389e8ab6108d1f62c9dcfe4afecefe16" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.69ex; height:2.176ex;" alt="{\displaystyle U=0.}"> Then the Lagrangian is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\left(\phi ,{\dot {\phi }}\right)=T-U={\tfrac {1}{2}}mr^{2}{\dot {\phi }}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ϕ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>T</mi> <mo>−</mo> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\left(\phi ,{\dot {\phi }}\right)=T-U={\tfrac {1}{2}}mr^{2}{\dot {\phi }}^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d732371c1f6d406cf5e23f94429c53b4d79c4474" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.079ex; height:4.843ex;" alt="{\displaystyle {\mathcal {L}}\left(\phi ,{\dot {\phi }}\right)=T-U={\tfrac {1}{2}}mr^{2}{\dot {\phi }}^{2}.}"> The generalized momentum "canonically conjugate to" the coordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\phi }={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}}}=mr^{2}{\dot {\phi }}=I\omega =L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>I</mi> <mi>ω</mi> <mo>=</mo> <mi>L</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\phi }={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}}}=mr^{2}{\dot {\phi }}=I\omega =L.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95f3e4e839968f840bd6e141c81ab3bbcebf7830" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:29.08ex; height:6.509ex;" alt="{\displaystyle p_{\phi }={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}}}=mr^{2}{\dot {\phi }}=I\omega =L.}"> <div class="mw-heading mw-heading3"><h3 id="Orbital_angular_momentum_in_three_dimensions">Orbital angular momentum in three dimensions</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=5" title="Edit section: Orbital angular momentum in three dimensions">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Torque_animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/09/Torque_animation.gif" decoding="async" width="220" height="154" class="mw-file-element" data-file-width="220" data-file-height="154" /></a><figcaption>Relationship between <a href="/wiki/Force" title="Force">force</a> (F), <a href="/wiki/Torque" title="Torque">torque</a> (τ), <a href="/wiki/Momentum" title="Momentum">momentum</a> (p), and angular momentum (L) vectors in a rotating system. r is the <a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position vector</a>.</figcaption></figure> To completely define orbital angular momentum in <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three dimensions</a>, it is required to know the rate at which the position vector sweeps out angle, the direction perpendicular to the instantaneous plane of angular displacement, and the <a href="/wiki/Mass" title="Mass">mass</a> involved, as well as how this mass is distributed in space.<a href="#cite_note-9">[9]</a> By retaining this <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional <a href="/wiki/Motion_(physics)" class="mw-redirect" title="Motion (physics)">motion</a> about the center of rotation – <a href="/wiki/Circular_motion" title="Circular motion">circular</a>, <a href="/wiki/Linear_motion" title="Linear motion">linear</a>, or otherwise. In <a href="/wiki/Vector_notation" title="Vector notation">vector notation</a>, the orbital angular momentum of a <a href="/wiki/Point_particle" title="Point particle">point particle</a> in motion about the origin can be expressed as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb18c91ec4284e8e4142e3b7fea1b3cb54baf40" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.194ex; height:2.509ex;" alt="{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},}"> where <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=r^{2}m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=r^{2}m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce8ff961fab0cdfe7ac50e8bee6d6ef1ee62347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;" alt="{\displaystyle I=r^{2}m}"> is the <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> for a <a href="/wiki/Point_particle" title="Point particle">point mass</a>,</li> <li><a href="/w/index.php?title=Special:MathWikibase&qid=Q161635" style="color:inherit;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}={\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}={\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d106050e9a58068eb301336636fb3aae9ac540f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.957ex; height:5.176ex;" alt="{\displaystyle {\boldsymbol {\omega }}={\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}}"></a> is the orbital <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> of the particle about the origin,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> is the position vector of the particle relative to the origin, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=\left\vert \mathbf {r} \right\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\left\vert \mathbf {r} \right\vert }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188bf31840a7fed7f6b36f9c107664c5881a7969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.543ex; height:2.843ex;" alt="{\displaystyle r=\left\vert \mathbf {r} \right\vert }">,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> is the linear velocity of the particle relative to the origin, and</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> is the mass of the particle.</li></ul> This can be expanded, reduced, and by the rules of <a href="/wiki/Vector_calculus" title="Vector calculus">vector algebra</a>, rearranged: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=\left(r^{2}m\right)\left({\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}\right)\\&=m\left(\mathbf {r} \times \mathbf {v} \right)\\&=\mathbf {r} \times m\mathbf {v} \\&=\mathbf {r} \times \mathbf {p} ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {L} &=\left(r^{2}m\right)\left({\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}\right)\\&=m\left(\mathbf {r} \times \mathbf {v} \right)\\&=\mathbf {r} \times m\mathbf {v} \\&=\mathbf {r} \times \mathbf {p} ,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a06879332a77f41eab2433dce133a996b34e0716" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:21.729ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\left(r^{2}m\right)\left({\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}\right)\\&=m\left(\mathbf {r} \times \mathbf {v} \right)\\&=\mathbf {r} \times m\mathbf {v} \\&=\mathbf {r} \times \mathbf {p} ,\end{aligned}}}"> which is the <a href="/wiki/Cross_product" title="Cross product">cross product</a> of the position vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> and the linear momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =m\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =m\mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a271a96e7b925fd39686375167c76d406e87c813" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.035ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} =m\mathbf {v} }"> of the particle. By the definition of the cross product, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5f750865376a1a4ae2b15a00b4ff9c75a66630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.608ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} }"> vector is <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }">. It is directed perpendicular to the plane of angular displacement, as indicated by the <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand rule</a> – so that the angular velocity is seen as <a href="/wiki/Clockwise" title="Clockwise">counter-clockwise</a> from the head of the vector. Conversely, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5f750865376a1a4ae2b15a00b4ff9c75a66630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.608ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} }"> vector defines the <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"> lie. By defining a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {u}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">u</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {u}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1adccb18b3b18c193af9f9ca2b0c0c500103e3ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {u}} }"> perpendicular to the plane of angular displacement, a <a href="/wiki/Angular_frequency" title="Angular frequency">scalar angular speed</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> results, where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \mathbf {\hat {u}} ={\boldsymbol {\omega }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">u</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \mathbf {\hat {u}} ={\boldsymbol {\omega }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc410d6366c23541fe0100e618a54d8f293a6b55" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.345ex; height:2.676ex;" alt="{\displaystyle \omega \mathbf {\hat {u}} ={\boldsymbol {\omega }},}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\frac {v_{\perp }}{r}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {v_{\perp }}{r}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c29a5398be29c77b66df211e844987301487fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.666ex; height:4.676ex;" alt="{\displaystyle \omega ={\frac {v_{\perp }}{r}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\perp }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\perp }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f80a8cf80254aa3ef2640555e94986487d5cba0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.638ex; height:2.009ex;" alt="{\displaystyle v_{\perp }}"> is the perpendicular component of the motion, as above. The two-dimensional scalar equations of the previous section can thus be given direction: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=I{\boldsymbol {\omega }}\\&=I\omega \mathbf {\hat {u}} \\&=\left(r^{2}m\right)\omega \mathbf {\hat {u}} \\&=rmv_{\perp }\mathbf {\hat {u}} \\&=r_{\perp }mv\mathbf {\hat {u}} ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>I</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">u</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">u</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>m</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">u</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> <mi>m</mi> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">u</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {L} &=I{\boldsymbol {\omega }}\\&=I\omega \mathbf {\hat {u}} \\&=\left(r^{2}m\right)\omega \mathbf {\hat {u}} \\&=rmv_{\perp }\mathbf {\hat {u}} \\&=r_{\perp }mv\mathbf {\hat {u}} ,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/138200e265424b4778f846322c7d77ae28571d5e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:15.05ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {L} &=I{\boldsymbol {\omega }}\\&=I\omega \mathbf {\hat {u}} \\&=\left(r^{2}m\right)\omega \mathbf {\hat {u}} \\&=rmv_{\perp }\mathbf {\hat {u}} \\&=r_{\perp }mv\mathbf {\hat {u}} ,\end{aligned}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =rmv\mathbf {\hat {u}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mi>r</mi> <mi>m</mi> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">u</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =rmv\mathbf {\hat {u}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a761fad402cde98517baffffd37adae166f765d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.409ex; height:2.343ex;" alt="{\displaystyle \mathbf {L} =rmv\mathbf {\hat {u}} }"> for circular motion, where all of the motion is perpendicular to the radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">. In the <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinate system</a> the angular momentum vector expresses as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =m\mathbf {r} \times \mathbf {v} =mr^{2}\left({\dot {\theta }}\,{\hat {\boldsymbol {\varphi }}}-{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\theta }}} \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">φ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =m\mathbf {r} \times \mathbf {v} =mr^{2}\left({\dot {\theta }}\,{\hat {\boldsymbol {\varphi }}}-{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\theta }}} \right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95551fade90d8e24bb3588cb75aa703a8d8e843e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.06ex; height:4.843ex;" alt="{\displaystyle \mathbf {L} =m\mathbf {r} \times \mathbf {v} =mr^{2}\left({\dot {\theta }}\,{\hat {\boldsymbol {\varphi }}}-{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\theta }}} \right).}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Analogy_to_linear_momentum">Analogy to linear momentum</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=6" title="Edit section: Analogy to linear momentum">edit</a>]</div> Angular momentum can be described as the rotational analog of <a href="/wiki/Linear_momentum" class="mw-redirect" title="Linear momentum">linear momentum</a>. Like linear momentum it involves elements of <a href="/wiki/Mass" title="Mass">mass</a> and <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a>. Unlike linear momentum it also involves elements of <a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position</a> and <a href="/wiki/Shape" title="Shape">shape</a>. Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? <a href="/wiki/Energy" title="Energy">Energy</a>, the ability to do <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>, can be stored in matter by setting it in motion—a combination of its <a href="/wiki/Inertia" title="Inertia">inertia</a> and its displacement. Inertia is measured by its <a href="/wiki/Mass" title="Mass">mass</a>, and displacement by its <a href="/wiki/Velocity" title="Velocity">velocity</a>. Their product, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{amount of (inertia⋅displacement)}}\\{\text{mass}}\times {\text{velocity}}&={\text{momentum}}\\m\times v&=p\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>amount of inertia</mtext> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>amount of displacement</mtext> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>amount of (inertia⋅displacement)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>mass</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>velocity</mtext> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>momentum</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> <mo>×</mo> <mi>v</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{amount of (inertia⋅displacement)}}\\{\text{mass}}\times {\text{velocity}}&={\text{momentum}}\\m\times v&=p\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0cb2452f95a0f974572985f205329a1d55bc71" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:84.693ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{amount of (inertia⋅displacement)}}\\{\text{mass}}\times {\text{velocity}}&={\text{momentum}}\\m\times v&=p\\\end{aligned}}}"> is the matter's <a href="/wiki/Momentum" title="Momentum">momentum</a>.<a href="#cite_note-10">[10]</a> Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a <a href="/wiki/Lever" title="Lever">lever</a> of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the moment arm. It has the effect of multiplying the momentum's effort in proportion to its length, an effect known as a <a href="/wiki/Moment_(physics)" title="Moment (physics)">moment.</a> Hence, the particle's momentum referred to a particular point, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}({\text{moment arm}})\times ({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{moment of (inertia⋅displacement)}}\\{\text{length}}\times {\text{mass}}\times {\text{velocity}}&={\text{moment of momentum}}\\r\times m\times v&=L\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>moment arm</mtext> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>amount of inertia</mtext> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>amount of displacement</mtext> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>moment of (inertia⋅displacement)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>length</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mass</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>velocity</mtext> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>moment of momentum</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mo>×</mo> <mi>m</mi> <mo>×</mo> <mi>v</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>L</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}({\text{moment arm}})\times ({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{moment of (inertia⋅displacement)}}\\{\text{length}}\times {\text{mass}}\times {\text{velocity}}&={\text{moment of momentum}}\\r\times m\times v&=L\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4002dc08fbcabcd33066cf654e7fd49e1adc402e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:102.711ex; height:9.009ex;" alt="{\displaystyle {\begin{aligned}({\text{moment arm}})\times ({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{moment of (inertia⋅displacement)}}\\{\text{length}}\times {\text{mass}}\times {\text{velocity}}&={\text{moment of momentum}}\\r\times m\times v&=L\\\end{aligned}}}"> is the angular momentum, sometimes called, as here, the moment of momentum of the particle versus that particular center point. The equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=rmv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>r</mi> <mi>m</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=rmv}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cae824c40f5d6282d0a4d286c1bdb7a7382c925e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.898ex; height:2.176ex;" alt="{\displaystyle L=rmv}"> combines a moment (a mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> turning moment arm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">) with a linear (straight-line equivalent) speed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}">. Linear speed referred to the central point is simply the product of the distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> and the angular speed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> versus the point: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=r\omega ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>r</mi> <mi>ω</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=r\omega ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574eabd58d8470d1f1eded75c6b545fb35f48d5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.367ex; height:2.009ex;" alt="{\displaystyle v=r\omega ,}"> another moment. Hence, angular momentum contains a double moment: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=rmr\omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>r</mi> <mi>m</mi> <mi>r</mi> <mi>ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=rmr\omega .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e34e11145331181be5f931f586322cd5b0a81aee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.912ex; height:2.176ex;" alt="{\displaystyle L=rmr\omega .}"> Simplifying slightly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=r^{2}m\omega ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mi>ω</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=r^{2}m\omega ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d901cacb8e9e59ef9ac4d17913b6d7922c47808f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.917ex; height:3.009ex;" alt="{\displaystyle L=r^{2}m\omega ,}"> the quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bbbb15eaf46c7e295eefc1ceb32d90161f7bac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.143ex; height:2.676ex;" alt="{\displaystyle r^{2}m}"> is the particle's <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>, sometimes called the second moment of mass. It is a measure of rotational inertia.<a href="#cite_note-11">[11]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Moment_of_inertia_examples.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Moment_of_inertia_examples.gif/220px-Moment_of_inertia_examples.gif" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Moment_of_inertia_examples.gif/330px-Moment_of_inertia_examples.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Moment_of_inertia_examples.gif/440px-Moment_of_inertia_examples.gif 2x" data-file-width="640" data-file-height="480" /></a><figcaption><a href="/wiki/Moment_of_inertia" title="Moment of inertia">Moment of inertia</a> (shown here), and therefore angular momentum, is different for each shown configuration of <a href="/wiki/Mass" title="Mass">mass</a> and <a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">axis of rotation</a>.</figcaption></figure> The above analogy of the translational momentum and rotational momentum can be expressed in vector form:[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {p} =m\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {p} =m\mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc2e572b5c0a5b7cd86beb2652f17d86bfb79a7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.035ex; height:2.009ex;" alt="{\textstyle \mathbf {p} =m\mathbf {v} }"> for linear motion</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794f5746b0089a4b7de0494195195a6291eda0aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.547ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }}}"> for rotation</li></ul> The direction of momentum is related to the direction of the velocity for linear movement. The direction of angular momentum is related to the angular velocity of the rotation. Because <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the <a href="/wiki/Square_(algebra)" title="Square (algebra)">squares</a> of their <a href="/wiki/Distances" class="mw-redirect" title="Distances">distances</a> from the center of rotation.<a href="#cite_note-Oberg-12">[12]</a> Therefore, the total moment of inertia, and the angular momentum, is a complex function of the configuration of the <a href="/wiki/Matter" title="Matter">matter</a> about the center of rotation and the orientation of the rotation for the various bits. For a <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a>, for instance a wheel or an asteroid, the orientation of rotation is simply the position of the <a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">rotation axis</a> versus the matter of the body. It may or may not pass through the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a>, or it may lie completely outside of the body. For the same body, angular momentum may take a different value for every possible axis about which rotation may take place.<a href="#cite_note-13">[13]</a> It reaches a minimum when the axis passes through the center of mass.<a href="#cite_note-14">[14]</a> For a collection of objects revolving about a center, for instance all of the bodies of the <a href="/wiki/Solar_System" title="Solar System">Solar System</a>, the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to the system's axis. Their orientations may also be completely random. In brief, the more mass and the farther it is from the center of rotation (the longer the <a href="/wiki/Torque#Moment_arm_formula" title="Torque">moment arm</a>), the greater the moment of inertia, and therefore the greater the angular momentum for a given <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a>. In many cases the <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>, and hence the angular momentum, can be simplified by,<a href="#cite_note-15">[15]</a> <a href="/w/index.php?title=Special:MathWikibase&qid=Q165618" style="color:inherit;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=k^{2}m,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=k^{2}m,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9262e5be6a71693fea700ba53795be70fde6291c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.223ex; height:3.009ex;" alt="{\displaystyle I=k^{2}m,}"></a>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> is the <a href="/wiki/Radius_of_gyration" title="Radius of gyration">radius of gyration</a>, the distance from the axis at which the entire mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> may be considered as concentrated. Similarly, for a <a href="/wiki/Point_particle" title="Point particle">point mass</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> the <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> is defined as, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=r^{2}m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=r^{2}m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce8ff961fab0cdfe7ac50e8bee6d6ef1ee62347" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;" alt="{\displaystyle I=r^{2}m}">where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> is the <a href="/wiki/Radius" title="Radius">radius</a> of the point mass from the center of rotation, and for any collection of particles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec8e804f69706d3f5ad235f4f983220c8df7c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle m_{i}}"> as the sum, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}I_{i}=\sum _{i}r_{i}^{2}m_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}I_{i}=\sum _{i}r_{i}^{2}m_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e77d22473cf9495a86e27ee160fcfbd35b50253" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.348ex; height:5.509ex;" alt="{\displaystyle \sum _{i}I_{i}=\sum _{i}r_{i}^{2}m_{i}}"> Angular momentum's dependence on position and shape is reflected in its <a href="/wiki/Units_of_measurement" class="mw-redirect" title="Units of measurement">units</a> versus linear momentum: kg⋅m2/s or N⋅m⋅s for angular momentum versus <a href="/wiki/SI_derived_unit" title="SI derived unit">kg⋅m/s</a> or <a href="/wiki/SI_derived_unit" title="SI derived unit">N⋅s</a> for linear momentum. When calculating angular momentum as the product of the moment of inertia times the angular velocity, the angular velocity must be expressed in radians per second, where the radian assumes the dimensionless value of unity. (When performing dimensional analysis, it may be productive to use <a href="/wiki/Dimensional_analysis#Siano's_extension:_orientational_analysis" title="Dimensional analysis">orientational analysis</a> which treats radians as a base unit, but this is not done in the <a href="/wiki/International_system_of_units" class="mw-redirect" title="International system of units">International system of units</a>). The units if angular momentum can be interpreted as <a href="/wiki/Torque" title="Torque">torque</a>⋅time. An object with angular momentum of L N⋅m⋅s can be reduced to zero angular velocity by an angular <a href="/wiki/Impulse_(physics)" title="Impulse (physics)">impulse</a> of L N⋅m⋅s.<a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a> The <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to the axis of angular momentum and passing through the center of mass<a href="#cite_note-18">[18]</a> is sometimes called the invariable plane, because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered.<a href="#cite_note-Rankine-19">[19]</a> One such plane is the <a href="/wiki/Invariable_plane" title="Invariable plane">invariable plane of the Solar System</a>. <div class="mw-heading mw-heading3"><h3 id="Angular_momentum_and_torque">Angular momentum and torque</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=7" title="Edit section: Angular momentum and torque">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Torque#Relationship_with_the_angular_momentum" title="Torque">Torque § Relationship with the angular momentum</a></div> <a href="/wiki/Newton%27s_laws_of_motion#Newton's_second_law" title="Newton's laws of motion">Newton's second law of motion</a> can be expressed mathematically, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =m\mathbf {a} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m\mathbf {a} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93aa54e6c7e8df66d85d06b6eb0b0a2d3ec4ce20" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.768ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =m\mathbf {a} ,}"> or <a href="/wiki/Force" title="Force">force</a> = <a href="/wiki/Mass" title="Mass">mass</a> × <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>. The rotational equivalent for point particles may be derived as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794f5746b0089a4b7de0494195195a6291eda0aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.547ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }}}"> which means that the torque (i.e. the time <a href="/wiki/Derivative" title="Derivative">derivative</a> of the angular momentum) is <a href="/w/index.php?title=Special:MathWikibase&qid=Q48103" style="color:inherit;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}={\frac {dI}{dt}}{\boldsymbol {\omega }}+I{\frac {d{\boldsymbol {\omega }}}{dt}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>I</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>+</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}={\frac {dI}{dt}}{\boldsymbol {\omega }}+I{\frac {d{\boldsymbol {\omega }}}{dt}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c66790aae79872ce38c8f57bb7666a5e5bf05b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.789ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {\tau }}={\frac {dI}{dt}}{\boldsymbol {\omega }}+I{\frac {d{\boldsymbol {\omega }}}{dt}}.}"></a> Because the moment of inertia is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mr^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mr^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.143ex; height:2.676ex;" alt="{\displaystyle mr^{2}}">, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dI}{dt}}=2mr{\frac {dr}{dt}}=2rp_{||}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>I</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>m</mi> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>r</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dI}{dt}}=2mr{\frac {dr}{dt}}=2rp_{||}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7393f1bd48479c3e727ed9bd2a44ba8959f28e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.3ex; height:5.509ex;" alt="{\displaystyle {\frac {dI}{dt}}=2mr{\frac {dr}{dt}}=2rp_{||}}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\mathbf {L} }{dt}}=I{\frac {d{\boldsymbol {\omega }}}{dt}}+2rp_{||}{\boldsymbol {\omega }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>r</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {L} }{dt}}=I{\frac {d{\boldsymbol {\omega }}}{dt}}+2rp_{||}{\boldsymbol {\omega }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fbdc5103e70d211066c7dc4abe45e04df273953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.335ex; height:5.509ex;" alt="{\displaystyle {\frac {d\mathbf {L} }{dt}}=I{\frac {d{\boldsymbol {\omega }}}{dt}}+2rp_{||}{\boldsymbol {\omega }},}"> which, reduces to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }}+2rp_{||}{\boldsymbol {\omega }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <mo>=</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> <mo>+</mo> <mn>2</mn> <mi>r</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }}+2rp_{||}{\boldsymbol {\omega }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf36dafd07a96cefd7b472c30ea9b6e1dd4d3bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.14ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }}+2rp_{||}{\boldsymbol {\omega }}.}"> This is the rotational analog of Newton's second law. Note that the torque is not necessarily proportional or parallel to the angular acceleration (as one might expect). The reason for this is that the moment of inertia of a particle can change with time, something that cannot occur for ordinary mass. <div class="mw-heading mw-heading2"><h2 id="Conservation_of_angular_momentum">Conservation of angular momentum</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=8" title="Edit section: Conservation of angular momentum">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cup_of_Russia_2010_-_Yuko_Kawaguti_(2).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg/170px-Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg" decoding="async" width="170" height="277" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg/255px-Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/6/68/Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg 2x" data-file-width="307" data-file-height="500" /></a><figcaption>A <a href="/wiki/Figure_skating" title="Figure skating">figure skater</a> in a spin uses conservation of angular momentum – decreasing her <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> by drawing in her arms and legs increases her <a href="/wiki/Angular_velocity" title="Angular velocity">rotational speed</a>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="General_considerations">General considerations</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=9" title="Edit section: General considerations">edit</a>]</div> A rotational analog of <a href="/wiki/Newton%27s_laws_of_motion#Newton's_third_law" title="Newton's laws of motion">Newton's third law of motion</a> might be written, "In a <a href="/wiki/Closed_system" title="Closed system">closed system</a>, no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque about the same axis."<a href="#cite_note-Crew-20">[20]</a> Hence, angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved).<a href="#cite_note-21">[21]</a> Seen another way, a rotational analogue of <a href="/wiki/Newton%27s_laws_of_motion#Newton's_first_law" title="Newton's laws of motion">Newton's first law of motion</a> might be written, "A rigid body continues in a state of uniform rotation unless acted upon by an external influence."<a href="#cite_note-Crew-20">[20]</a> Thus with no external influence to act upon it, the original angular momentum of the system remains constant.<a href="#cite_note-22">[22]</a> The conservation of angular momentum is used in analyzing <a href="/wiki/Classical_central-force_problem" title="Classical central-force problem">central force motion</a>. If the net force on some body is directed always toward some point, the center, then there is no torque on the body with respect to the center, as all of the force is directed along the <a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">radius vector</a>, and none is <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to the radius. Mathematically, torque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} =\mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} =\mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce23213f7078a404609088e6b314db820290527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.224ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} =\mathbf {0} ,}"> because in this case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da18bef8c979f3548bb0d8976f5844012d7b8256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }"> are parallel vectors. Therefore, the angular momentum of the body about the center is constant. This is the case with <a href="/wiki/Gravity" title="Gravity">gravitational attraction</a> in the <a href="/wiki/Orbit" title="Orbit">orbits</a> of <a href="/wiki/Planet" title="Planet">planets</a> and <a href="/wiki/Satellite" title="Satellite">satellites</a>, where the gravitational force is always directed toward the primary body and orbiting bodies conserve angular momentum by exchanging distance and velocity as they move about the primary. Central force motion is also used in the analysis of the <a href="/wiki/Bohr_model" title="Bohr model">Bohr model</a> of the <a href="/wiki/Atom" title="Atom">atom</a>. For a planet, angular momentum is distributed between the spin of the planet and its revolution in its orbit, and these are often exchanged by various mechanisms. The conservation of angular momentum in the <a href="/wiki/Lunar_theory" title="Lunar theory">Earth–Moon system</a> results in the transfer of angular momentum from Earth to Moon, due to <a href="/wiki/Tidal_acceleration" title="Tidal acceleration">tidal torque</a> the Moon exerts on the Earth. This in turn results in the slowing down of the rotation rate of Earth, at about 65.7 nanoseconds per day,<a href="#cite_note-23">[23]</a> and in gradual increase of the radius of Moon's orbit, at about 3.82 centimeters per year.<a href="#cite_note-24">[24]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PrecessionOfATop.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/PrecessionOfATop.svg/220px-PrecessionOfATop.svg.png" decoding="async" width="220" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/PrecessionOfATop.svg/330px-PrecessionOfATop.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/PrecessionOfATop.svg/440px-PrecessionOfATop.svg.png 2x" data-file-width="300" data-file-height="350" /></a><figcaption>The <a href="/wiki/Torque" title="Torque">torque</a> caused by the two opposing forces Fg and −Fg causes a change in the angular momentum L in the direction of that torque (since torque is the time derivative of angular momentum). This causes the <a href="/wiki/Spinning_top" title="Spinning top">top</a> to <a href="/wiki/Precess" class="mw-redirect" title="Precess">precess</a>.</figcaption></figure> The conservation of angular momentum explains the angular acceleration of an <a href="/wiki/Ice_skating" title="Ice skating">ice skater</a> as they bring their arms and legs close to the vertical axis of rotation. By bringing part of the mass of their body closer to the axis, they decrease their body's moment of inertia. Because angular momentum is the product of <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> and <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a>, if the angular momentum remains constant (is conserved), then the angular velocity (rotational speed) of the skater must increase. The same phenomenon results in extremely fast spin of compact stars (like <a href="/wiki/White_dwarf" title="White dwarf">white dwarfs</a>, <a href="/wiki/Neutron_star" title="Neutron star">neutron stars</a> and <a href="/wiki/Black_hole" title="Black hole">black holes</a>) when they are formed out of much larger and slower rotating stars. Conservation is not always a full explanation for the dynamics of a system but is a key constraint. For example, a <a href="/wiki/Spinning_top" title="Spinning top">spinning top</a> is subject to gravitational torque making it lean over and change the angular momentum about the <a href="/wiki/Nutation" title="Nutation">nutation</a> axis, but neglecting friction at the point of spinning contact, it has a conserved angular momentum about its spinning axis, and another about its <a href="/wiki/Precession" title="Precession">precession</a> axis. Also, in any <a href="/wiki/Planetary_system" title="Planetary system">planetary system</a>, the planets, star(s), comets, and asteroids can all move in numerous complicated ways, but only so that the angular momentum of the system is conserved. <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a> states that every <a href="/wiki/Conservation_law" title="Conservation law">conservation law</a> is associated with a <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> (invariant) of the underlying physics. The symmetry associated with conservation of angular momentum is <a href="/wiki/Rotational_invariance" title="Rotational invariance">rotational invariance</a>. The fact that the physics of a system is unchanged if it is rotated by any angle about an axis implies that angular momentum is conserved.<a href="#cite_note-25">[25]</a> <div class="mw-heading mw-heading3"><h3 id="Relation_to_Newton's_second_law_of_motion">Relation to Newton's second law of motion</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=10" title="Edit section: Relation to Newton's second law of motion">edit</a>]</div> While angular momentum total conservation can be understood separately from <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> as stemming from <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a> in systems symmetric under rotations, it can also be understood simply as an efficient method of calculation of results that can also be otherwise arrived at directly from Newton's second law, together with laws governing the forces of nature (such as Newton's third law, <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> and <a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a>). Indeed, given initial conditions of position and velocity for every point, and the forces at such a condition, one may use Newton's second law to calculate the second derivative of position, and solving for this gives full information on the development of the physical system with time.<a href="#cite_note-26">[26]</a> Note, however, that this is no longer true in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, due to the existence of <a href="/wiki/Spin_(physics)" title="Spin (physics)">particle spin</a>, which is angular momentum that cannot be described by the cumulative effect of point-like motions in space. As an example, consider decreasing of the <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>, e.g. when a <a href="/wiki/Figure_skating" title="Figure skating">figure skater</a> is pulling in their hands, speeding up the circular motion. In terms of angular momentum conservation, we have, for angular momentum L, moment of inertia I and angular velocity ω: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=dL=d(I\cdot \omega )=dI\cdot \omega +I\cdot d\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mi>d</mi> <mi>L</mi> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo>⋅</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mi>I</mi> <mo>⋅</mo> <mi>ω</mi> <mo>+</mo> <mi>I</mi> <mo>⋅</mo> <mi>d</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=dL=d(I\cdot \omega )=dI\cdot \omega +I\cdot d\omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15093607ae440e94bbf63e759eb62b687ab1110" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.444ex; height:2.843ex;" alt="{\displaystyle 0=dL=d(I\cdot \omega )=dI\cdot \omega +I\cdot d\omega }"> Using this, we see that the change requires an energy of: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dE=d\left({\tfrac {1}{2}}I\cdot \omega ^{2}\right)={\tfrac {1}{2}}dI\cdot \omega ^{2}+I\cdot \omega \cdot d\omega =-{\tfrac {1}{2}}dI\cdot \omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>E</mi> <mo>=</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>I</mi> <mo>⋅</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>d</mi> <mi>I</mi> <mo>⋅</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>I</mi> <mo>⋅</mo> <mi>ω</mi> <mo>⋅</mo> <mi>d</mi> <mi>ω</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>d</mi> <mi>I</mi> <mo>⋅</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dE=d\left({\tfrac {1}{2}}I\cdot \omega ^{2}\right)={\tfrac {1}{2}}dI\cdot \omega ^{2}+I\cdot \omega \cdot d\omega =-{\tfrac {1}{2}}dI\cdot \omega ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b138f0d46df7e9572073f9ffbaf30371999a41" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:52.764ex; height:3.509ex;" alt="{\displaystyle dE=d\left({\tfrac {1}{2}}I\cdot \omega ^{2}\right)={\tfrac {1}{2}}dI\cdot \omega ^{2}+I\cdot \omega \cdot d\omega =-{\tfrac {1}{2}}dI\cdot \omega ^{2}}"> so that a decrease in the moment of inertia requires investing energy. This can be compared to the work done as calculated using Newton's laws. Each point in the rotating body is accelerating, at each point of time, with radial acceleration of: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -r\cdot \omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>r</mi> <mo>⋅</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -r\cdot \omega ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/988f476c4a01afb3d9d6b6d6aba262632eeebea9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.036ex; height:2.843ex;" alt="{\displaystyle -r\cdot \omega ^{2}}"> Let us observe a point of mass m, whose position vector relative to the center of motion is perpendicular to the z-axis at a given point of time, and is at a distance z. The <a href="/wiki/Centripetal_force" title="Centripetal force">centripetal force</a> on this point, keeping the circular motion, is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -m\cdot z\cdot \omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>m</mi> <mo>⋅</mo> <mi>z</mi> <mo>⋅</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -m\cdot z\cdot \omega ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b5d5e9aee971b197b897a9c6f8c1e41968260d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.795ex; height:2.843ex;" alt="{\displaystyle -m\cdot z\cdot \omega ^{2}}"> Thus the work required for moving this point to a distance dz farther from the center of motion is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dW=-m\cdot z\cdot \omega ^{2}\cdot dz=-m\cdot \omega ^{2}\cdot d\left({\tfrac {1}{2}}z^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mo>⋅</mo> <mi>z</mi> <mo>⋅</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mo>⋅</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dW=-m\cdot z\cdot \omega ^{2}\cdot dz=-m\cdot \omega ^{2}\cdot d\left({\tfrac {1}{2}}z^{2}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4cb725b4a5fa01176b14940347a963885953f73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.868ex; height:3.509ex;" alt="{\displaystyle dW=-m\cdot z\cdot \omega ^{2}\cdot dz=-m\cdot \omega ^{2}\cdot d\left({\tfrac {1}{2}}z^{2}\right)}"> For a non-pointlike body one must integrate over this, with m replaced by the mass density per unit z. This gives: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dW=-{\tfrac {1}{2}}dI\cdot \omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>d</mi> <mi>I</mi> <mo>⋅</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dW=-{\tfrac {1}{2}}dI\cdot \omega ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56874f3287508ee23fb1f2afb9d94eef5c4b2eea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.783ex; height:3.509ex;" alt="{\displaystyle dW=-{\tfrac {1}{2}}dI\cdot \omega ^{2}}"> which is exactly the energy required for keeping the angular momentum conserved. Note, that the above calculation can also be performed per mass, using <a href="/wiki/Kinematics" title="Kinematics">kinematics</a> only. Thus the phenomena of figure skater accelerating tangential velocity while pulling their hands in, can be understood as follows in layman's language: The skater's palms are not moving in a straight line, so they are constantly accelerating inwards, but do not gain additional speed because the accelerating is always done when their motion inwards is zero. However, this is different when pulling the palms closer to the body: The acceleration due to rotation now increases the speed; but because of the rotation, the increase in speed does not translate to a significant speed inwards, but to an increase of the rotation speed. <div class="mw-heading mw-heading3"><h3 id="Stationary-action_principle">Stationary-action principle</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=11" title="Edit section: Stationary-action principle">edit</a>]</div> In classical mechanics it can be shown that the rotational invariance of action functionals implies conservation of angular momentum. The action is defined in classical physics as a functional of positions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644229f2cfb7cba7444d056ec5176b0d1ab7fc46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.778ex; height:2.843ex;" alt="{\displaystyle x_{i}(t)}"> often represented by the use of square brackets, and the final and initial times. It assumes the following form in cartesian coordinates:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\left([x_{i}];t_{1},t_{2}\right)\equiv \int _{t_{1}}^{t_{2}}dt\left({\frac {1}{2}}m{\frac {dx_{i}}{dt}}\ {\frac {dx_{i}}{dt}}-V(x_{i})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>;</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>≡</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\left([x_{i}];t_{1},t_{2}\right)\equiv \int _{t_{1}}^{t_{2}}dt\left({\frac {1}{2}}m{\frac {dx_{i}}{dt}}\ {\frac {dx_{i}}{dt}}-V(x_{i})\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a1a0a77886ae741be2679b9b7123e90de2e3ae9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.994ex; height:6.509ex;" alt="{\displaystyle S\left([x_{i}];t_{1},t_{2}\right)\equiv \int _{t_{1}}^{t_{2}}dt\left({\frac {1}{2}}m{\frac {dx_{i}}{dt}}\ {\frac {dx_{i}}{dt}}-V(x_{i})\right)}">where the repeated indices indicate summation over the index. If the action is invariant of an infinitesimal transformation, it can be mathematically stated as: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \delta S=S\left([x_{i}+\delta x_{i}];t_{1},t_{2}\right)-S\left([x_{i}];t_{1},t_{2}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>δ</mi> <mi>S</mi> <mo>=</mo> <mi>S</mi> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>;</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>S</mi> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>;</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \delta S=S\left([x_{i}+\delta x_{i}];t_{1},t_{2}\right)-S\left([x_{i}];t_{1},t_{2}\right)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/560e494d7450ac05174984f59efb1377447616e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.715ex; height:2.843ex;" alt="{\textstyle \delta S=S\left([x_{i}+\delta x_{i}];t_{1},t_{2}\right)-S\left([x_{i}];t_{1},t_{2}\right)=0}">. Under the transformation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\rightarrow x_{i}+\delta x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\rightarrow x_{i}+\delta x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10388124f5221ee2ff96225668b932b77a5f13fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.891ex; height:2.676ex;" alt="{\displaystyle x_{i}\rightarrow x_{i}+\delta x_{i}}">, the action becomes: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\left([x_{i}+\delta x_{i}];t_{1},t_{2}\right)=\!\int _{t_{1}}^{t_{2}}dt\left({\frac {1}{2}}m{\frac {d(x_{i}+\delta x_{i})}{dt}}{\frac {d(x_{i}+\delta x_{i})}{dt}}-V(x_{i}+\delta x_{i})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>;</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mspace width="negativethinmathspace" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\left([x_{i}+\delta x_{i}];t_{1},t_{2}\right)=\!\int _{t_{1}}^{t_{2}}dt\left({\frac {1}{2}}m{\frac {d(x_{i}+\delta x_{i})}{dt}}{\frac {d(x_{i}+\delta x_{i})}{dt}}-V(x_{i}+\delta x_{i})\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6cea9bc1b2c5c3fe5c29911ab7cedd3b088354b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:74.718ex; height:6.509ex;" alt="{\displaystyle S\left([x_{i}+\delta x_{i}];t_{1},t_{2}\right)=\!\int _{t_{1}}^{t_{2}}dt\left({\frac {1}{2}}m{\frac {d(x_{i}+\delta x_{i})}{dt}}{\frac {d(x_{i}+\delta x_{i})}{dt}}-V(x_{i}+\delta x_{i})\right)}"> where we can employ the expansion of the terms up-to first order in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \delta x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \delta x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0c1719d121852a8fd2910c6849d3c6c4339801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.178ex; height:2.676ex;" alt="{\textstyle \delta x_{i}}">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d(x_{i}+\delta x_{i})}{dt}}{\frac {d(x_{i}+\delta x_{i})}{dt}}&\simeq {\frac {dx_{i}}{dt}}{\frac {dx_{i}}{dt}}-2{\frac {d^{2}x_{i}}{dt^{2}}}\delta x_{i}+2{\frac {d}{dt}}\left(\delta x_{i}{\frac {dx_{i}}{dt}}\right)\\V(x_{i}+\delta x_{i})&\simeq V(x_{i})+\delta x_{i}{\frac {\partial V}{\partial x_{i}}}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>≃</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≃</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d(x_{i}+\delta x_{i})}{dt}}{\frac {d(x_{i}+\delta x_{i})}{dt}}&\simeq {\frac {dx_{i}}{dt}}{\frac {dx_{i}}{dt}}-2{\frac {d^{2}x_{i}}{dt^{2}}}\delta x_{i}+2{\frac {d}{dt}}\left(\delta x_{i}{\frac {dx_{i}}{dt}}\right)\\V(x_{i}+\delta x_{i})&\simeq V(x_{i})+\delta x_{i}{\frac {\partial V}{\partial x_{i}}}\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07663115c3170bd454b07a6aa838567be76673ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.294ex; margin-bottom: -0.211ex; width:66.712ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {d(x_{i}+\delta x_{i})}{dt}}{\frac {d(x_{i}+\delta x_{i})}{dt}}&\simeq {\frac {dx_{i}}{dt}}{\frac {dx_{i}}{dt}}-2{\frac {d^{2}x_{i}}{dt^{2}}}\delta x_{i}+2{\frac {d}{dt}}\left(\delta x_{i}{\frac {dx_{i}}{dt}}\right)\\V(x_{i}+\delta x_{i})&\simeq V(x_{i})+\delta x_{i}{\frac {\partial V}{\partial x_{i}}}\\\end{aligned}}}">giving the following change in action: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S[x_{i}+\delta x_{i}]\simeq S[x_{i}]+\int _{t_{1}}^{t_{2}}dt\,\delta x_{i}\left(-{\frac {\partial V}{\partial x_{i}}}-m{\frac {d^{2}x_{i}}{dt^{2}}}\right)+m\int _{t_{1}}^{t_{2}}dt{\frac {d}{dt}}\left(\delta x_{i}{\frac {dx_{i}}{dt}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>≃</mo> <mi>S</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>m</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[x_{i}+\delta x_{i}]\simeq S[x_{i}]+\int _{t_{1}}^{t_{2}}dt\,\delta x_{i}\left(-{\frac {\partial V}{\partial x_{i}}}-m{\frac {d^{2}x_{i}}{dt^{2}}}\right)+m\int _{t_{1}}^{t_{2}}dt{\frac {d}{dt}}\left(\delta x_{i}{\frac {dx_{i}}{dt}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d0a75604f66354be4b20123019ee46516b215e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:78.875ex; height:6.509ex;" alt="{\displaystyle S[x_{i}+\delta x_{i}]\simeq S[x_{i}]+\int _{t_{1}}^{t_{2}}dt\,\delta x_{i}\left(-{\frac {\partial V}{\partial x_{i}}}-m{\frac {d^{2}x_{i}}{dt^{2}}}\right)+m\int _{t_{1}}^{t_{2}}dt{\frac {d}{dt}}\left(\delta x_{i}{\frac {dx_{i}}{dt}}\right).}"> Since all rotations can be expressed as <a href="/wiki/3D_rotation_group#Exponential_map" title="3D rotation group">matrix exponential</a> of skew-symmetric matrices, i.e. as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R({\hat {n}},\theta )=e^{M\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R({\hat {n}},\theta )=e^{M\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2603ebefa9ea9e46f70503e6d179f5fc0902ccb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.005ex; height:3.176ex;" alt="{\displaystyle R({\hat {n}},\theta )=e^{M\theta }}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is a skew-symmetric matrix and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> is angle of rotation, we can express the change of coordinates due to the rotation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R({\hat {n}},\delta \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>δ</mi> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R({\hat {n}},\delta \theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bca338321baae64860660f620bb99d48e3678310" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.141ex; height:2.843ex;" alt="{\displaystyle R({\hat {n}},\delta \theta )}">, up-to first order of infinitesimal angle of rotation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bab72879c043fc87fd27b8b7098be098849695a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.139ex; height:2.343ex;" alt="{\displaystyle \delta \theta }"> as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x_{i}=M_{ij}x_{j}\delta \theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>δ</mi> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta x_{i}=M_{ij}x_{j}\delta \theta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dedeaa16d342d961468d421d2d38b4e7b4277b34" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.033ex; height:3.009ex;" alt="{\displaystyle \delta x_{i}=M_{ij}x_{j}\delta \theta .}"> Combining the equation of motion and rotational invariance of action, we get from the above equations that:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\delta S=\int _{t_{1}}^{t_{2}}dt{\frac {d}{dt}}\left(m{\frac {dx_{i}}{dt}}\delta x_{i}\right)=M_{ij}\,\delta \theta \,m\,x_{j}{\frac {dx_{i}}{dt}}{\Bigg \vert }_{t_{1}}^{t_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mi>δ</mi> <mi>S</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>m</mi> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\delta S=\int _{t_{1}}^{t_{2}}dt{\frac {d}{dt}}\left(m{\frac {dx_{i}}{dt}}\delta x_{i}\right)=M_{ij}\,\delta \theta \,m\,x_{j}{\frac {dx_{i}}{dt}}{\Bigg \vert }_{t_{1}}^{t_{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90376b22e55f2e7f198cea413901d1baac1faab3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.467ex; height:7.176ex;" alt="{\displaystyle 0=\delta S=\int _{t_{1}}^{t_{2}}dt{\frac {d}{dt}}\left(m{\frac {dx_{i}}{dt}}\delta x_{i}\right)=M_{ij}\,\delta \theta \,m\,x_{j}{\frac {dx_{i}}{dt}}{\Bigg \vert }_{t_{1}}^{t_{2}}}">Since this is true for any matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6136d919b3f10cc58cc023e05a5c36b2f44e111f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.731ex; height:2.843ex;" alt="{\displaystyle M_{ij}}"> that satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{ij}=-M_{ji},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{ij}=-M_{ji},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aaca2ed0e81e7e02541decbecba9a17c70cadc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.016ex; height:2.843ex;" alt="{\displaystyle M_{ij}=-M_{ji},}"> it results in the conservation of the following quantity: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{ij}(t):=m\left(x_{i}{\frac {dx_{j}}{dt}}-x_{j}{\frac {dx_{i}}{dt}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{ij}(t):=m\left(x_{i}{\frac {dx_{j}}{dt}}-x_{j}{\frac {dx_{i}}{dt}}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df700b150e2984a7eb4b75dac1e4a9f19497df1b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.406ex; height:6.343ex;" alt="{\displaystyle \ell _{ij}(t):=m\left(x_{i}{\frac {dx_{j}}{dt}}-x_{j}{\frac {dx_{i}}{dt}}\right),}"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{ij}(t_{1})=\ell _{ij}(t_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{ij}(t_{1})=\ell _{ij}(t_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49159037ee5d793aaff4612513371486a410b7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.398ex; height:3.009ex;" alt="{\displaystyle \ell _{ij}(t_{1})=\ell _{ij}(t_{2})}">. This corresponds to the conservation of angular momentum throughout the motion.<a href="#cite_note-27">[27]</a> <div class="mw-heading mw-heading3"><h3 id="Lagrangian_formalism">Lagrangian formalism</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=12" title="Edit section: Lagrangian formalism">edit</a>]</div> In <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>, angular momentum for rotation around a given axis, is the <a href="/wiki/Conjugate_momentum" class="mw-redirect" title="Conjugate momentum">conjugate momentum</a> of the <a href="/wiki/Generalized_coordinate" class="mw-redirect" title="Generalized coordinate">generalized coordinate</a> of the angle around the same axis. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a5b940110c5e1fe03782a31c5e700939ae20e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.585ex; height:2.509ex;" alt="{\displaystyle L_{z}}">, the angular momentum around the z axis, is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{z}={\frac {\partial {\cal {L}}}{\partial {\dot {\theta }}_{z}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{z}={\frac {\partial {\cal {L}}}{\partial {\dot {\theta }}_{z}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46ec2a2d5043dc0be22545107c74c2a5d9e703a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:10.195ex; height:6.343ex;" alt="{\displaystyle L_{z}={\frac {\partial {\cal {L}}}{\partial {\dot {\theta }}_{z}}}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cal {L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cddbb21ad79aa4e70f27927e433fd985873a3b6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\cal {L}}}"> is the Lagrangian and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/043cbe96ac7ea3ea664c287abee4ea1819373353" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.092ex; height:2.509ex;" alt="{\displaystyle \theta _{z}}"> is the angle around the z axis. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\theta }}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb1c138bd0e051666aae97e19177d0f45976d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:3.176ex;" alt="{\displaystyle {\dot {\theta }}_{z}}">, the time derivative of the angle, is the <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6664a519a737b4e06b4668df3e837d4e4986db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.447ex; height:2.009ex;" alt="{\displaystyle \omega _{z}}">. Ordinarily, the Lagrangian depends on the angular velocity through the kinetic energy: The latter can be written by separating the velocity to its radial and tangential part, with the tangential part at the x-y plane, around the z-axis, being equal to: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}{\tfrac {1}{2}}m_{i}{v_{T}}_{i}^{2}=\sum _{i}{\tfrac {1}{2}}m_{i}\left(x_{i}^{2}+y_{i}^{2}\right){{\omega _{z}}_{i}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}{\tfrac {1}{2}}m_{i}{v_{T}}_{i}^{2}=\sum _{i}{\tfrac {1}{2}}m_{i}\left(x_{i}^{2}+y_{i}^{2}\right){{\omega _{z}}_{i}}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/540dac61440dd8da1145e3c998d2f9562d2213f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.795ex; height:5.509ex;" alt="{\displaystyle \sum _{i}{\tfrac {1}{2}}m_{i}{v_{T}}_{i}^{2}=\sum _{i}{\tfrac {1}{2}}m_{i}\left(x_{i}^{2}+y_{i}^{2}\right){{\omega _{z}}_{i}}^{2}}"> where the subscript i stands for the i-th body, and m, vT and ωz stand for mass, tangential velocity around the z-axis and angular velocity around that axis, respectively. For a body that is not point-like, with density ρ, we have instead: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\int \rho (x,y,z)\left(x_{i}^{2}+y_{i}^{2}\right){{\omega _{z}}_{i}}^{2}\,dx\,dy={\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>∫</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\int \rho (x,y,z)\left(x_{i}^{2}+y_{i}^{2}\right){{\omega _{z}}_{i}}^{2}\,dx\,dy={\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ad9f064c3713ba6d5a5584eb634b562149b80" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.177ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{2}}\int \rho (x,y,z)\left(x_{i}^{2}+y_{i}^{2}\right){{\omega _{z}}_{i}}^{2}\,dx\,dy={\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}}"> where integration runs over the area of the body,<a href="#cite_note-28">[28]</a> and Iz is the moment of inertia around the z-axis. Thus, assuming the potential energy does not depend on ωz (this assumption may fail for electromagnetic systems), we have the angular momentum of the ith object: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{L_{z}}_{i}&={\frac {\partial {\cal {L}}}{\partial {{\omega _{z}}_{i}}}}={\frac {\partial E_{k}}{\partial {{\omega _{z}}_{i}}}}\\&={I_{z}}_{i}\cdot {\omega _{z}}_{i}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{L_{z}}_{i}&={\frac {\partial {\cal {L}}}{\partial {{\omega _{z}}_{i}}}}={\frac {\partial E_{k}}{\partial {{\omega _{z}}_{i}}}}\\&={I_{z}}_{i}\cdot {\omega _{z}}_{i}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced48449127d30dc5fcde4c5eab09328b68c93b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:21.135ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}{L_{z}}_{i}&={\frac {\partial {\cal {L}}}{\partial {{\omega _{z}}_{i}}}}={\frac {\partial E_{k}}{\partial {{\omega _{z}}_{i}}}}\\&={I_{z}}_{i}\cdot {\omega _{z}}_{i}\end{aligned}}}"> We have thus far rotated each object by a separate angle; we may also define an overall angle θz by which we rotate the whole system, thus rotating also each object around the z-axis, and have the overall angular momentum: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{z}=\sum _{i}{I_{z}}_{i}\cdot {\omega _{z}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{z}=\sum _{i}{I_{z}}_{i}\cdot {\omega _{z}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15e4a01264119c9ff46f2e4099dcc92eec496a41" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.176ex; height:5.509ex;" alt="{\displaystyle L_{z}=\sum _{i}{I_{z}}_{i}\cdot {\omega _{z}}_{i}}"> From <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equations</a> it then follows that: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d}{dt}}\left({\frac {\partial {\cal {L}}}{\partial {{{\dot {\theta }}_{z}}_{i}}}}\right)={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d{L_{z}}_{i}}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d}{dt}}\left({\frac {\partial {\cal {L}}}{\partial {{{\dot {\theta }}_{z}}_{i}}}}\right)={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d{L_{z}}_{i}}{dt}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eca9ec00bbbf83969b41733aaa004b16f23484" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.84ex; height:7.509ex;" alt="{\displaystyle 0={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d}{dt}}\left({\frac {\partial {\cal {L}}}{\partial {{{\dot {\theta }}_{z}}_{i}}}}\right)={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d{L_{z}}_{i}}{dt}}}"> Since the lagrangian is dependent upon the angles of the object only through the potential, we have: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d{L_{z}}_{i}}{dt}}={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}=-{\frac {\partial V}{\partial {{\theta _{z}}_{i}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d{L_{z}}_{i}}{dt}}={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}=-{\frac {\partial V}{\partial {{\theta _{z}}_{i}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e200f643c16a72c101334aea3c7b2fcb031059e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.533ex; height:6.009ex;" alt="{\displaystyle {\frac {d{L_{z}}_{i}}{dt}}={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}=-{\frac {\partial V}{\partial {{\theta _{z}}_{i}}}}}"> which is the torque on the ith object. Suppose the system is invariant to rotations, so that the potential is independent of an overall rotation by the angle θz (thus it may depend on the angles of objects only through their differences, in the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V({\theta _{z}}_{i},{\theta _{z}}_{j})=V({\theta _{z}}_{i}-{\theta _{z}}_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V({\theta _{z}}_{i},{\theta _{z}}_{j})=V({\theta _{z}}_{i}-{\theta _{z}}_{j})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf948ed90f5ec985945fc05ff41552e955811fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.953ex; height:3.009ex;" alt="{\displaystyle V({\theta _{z}}_{i},{\theta _{z}}_{j})=V({\theta _{z}}_{i}-{\theta _{z}}_{j})}">). We therefore get for the total angular momentum: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dL_{z}}{dt}}=-{\frac {\partial V}{\partial {\theta _{z}}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dL_{z}}{dt}}=-{\frac {\partial V}{\partial {\theta _{z}}}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aadf954d3dc9b7f87b433a5420606bc0e0fc27b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.05ex; height:5.843ex;" alt="{\displaystyle {\frac {dL_{z}}{dt}}=-{\frac {\partial V}{\partial {\theta _{z}}}}=0}"> And thus the angular momentum around the z-axis is conserved. This analysis can be repeated separately for each axis, giving conversation of the angular momentum vector. However, the angles around the three axes cannot be treated simultaneously as generalized coordinates, since they are not independent; in particular, two angles per point suffice to determine its position. While it is true that in the case of a rigid body, fully describing it requires, in addition to three <a href="/wiki/Translational_symmetry" title="Translational symmetry">translational</a> degrees of freedom, also specification of three rotational degrees of freedom; however these cannot be defined as rotations around the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian axes</a> (see <a href="/wiki/Euler_angles" title="Euler angles">Euler angles</a>). This caveat is reflected in quantum mechanics in the non-trivial <a href="/wiki/Commutation_relation" class="mw-redirect" title="Commutation relation">commutation relations</a> of the different components of the <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">angular momentum operator</a>. <div class="mw-heading mw-heading3"><h3 id="Hamiltonian_formalism">Hamiltonian formalism</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=13" title="Edit section: Hamiltonian formalism">edit</a>]</div> Equivalently, in <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a> the Hamiltonian can be described as a function of the angular momentum. As before, the part of the kinetic energy related to rotation around the z-axis for the ith object is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}={\frac {{{L_{z}}_{i}}^{2}}{2{I_{z}}_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}={\frac {{{L_{z}}_{i}}^{2}}{2{I_{z}}_{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fea4670d6d67a9c53d3b14e287afe4b053b0de0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.498ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}={\frac {{{L_{z}}_{i}}^{2}}{2{I_{z}}_{i}}}}"> which is analogous to the energy dependence upon momentum along the z-axis, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{{p_{z}}_{i}}^{2}}{{2m}_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{{p_{z}}_{i}}^{2}}{{2m}_{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a130e7f0e2bb096205ed92293b706638b0422cb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:4.861ex; height:6.176ex;" alt="{\displaystyle {\frac {{{p_{z}}_{i}}^{2}}{{2m}_{i}}}}">. Hamilton's equations relate the angle around the z-axis to its conjugate momentum, the angular momentum around the same axis: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d{\theta _{z}}_{i}}{dt}}&={\frac {\partial {\mathcal {H}}}{\partial {L_{z}}_{i}}}={\frac {{L_{z}}_{i}}{{I_{z}}_{i}}}\\{\frac {d{L_{z}}_{i}}{dt}}&=-{\frac {\partial {\mathcal {H}}}{\partial {\theta _{z}}_{i}}}=-{\frac {\partial V}{\partial {\theta _{z}}_{i}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d{\theta _{z}}_{i}}{dt}}&={\frac {\partial {\mathcal {H}}}{\partial {L_{z}}_{i}}}={\frac {{L_{z}}_{i}}{{I_{z}}_{i}}}\\{\frac {d{L_{z}}_{i}}{dt}}&=-{\frac {\partial {\mathcal {H}}}{\partial {\theta _{z}}_{i}}}=-{\frac {\partial V}{\partial {\theta _{z}}_{i}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fffb1fe3d60ae0a0ab90e9f3b2556e026b5053f0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:26.093ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {d{\theta _{z}}_{i}}{dt}}&={\frac {\partial {\mathcal {H}}}{\partial {L_{z}}_{i}}}={\frac {{L_{z}}_{i}}{{I_{z}}_{i}}}\\{\frac {d{L_{z}}_{i}}{dt}}&=-{\frac {\partial {\mathcal {H}}}{\partial {\theta _{z}}_{i}}}=-{\frac {\partial V}{\partial {\theta _{z}}_{i}}}\end{aligned}}}"> The first equation gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {L_{z}}_{i}={I_{z}}_{i}\cdot {{{\dot {\theta }}_{z}}_{i}}={I_{z}}_{i}\cdot {\omega _{z}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {L_{z}}_{i}={I_{z}}_{i}\cdot {{{\dot {\theta }}_{z}}_{i}}={I_{z}}_{i}\cdot {\omega _{z}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fff0339cc32d6f14853026e32bfeaa291018302" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.993ex; height:3.176ex;" alt="{\displaystyle {L_{z}}_{i}={I_{z}}_{i}\cdot {{{\dot {\theta }}_{z}}_{i}}={I_{z}}_{i}\cdot {\omega _{z}}_{i}}"> And so we get the same results as in the Lagrangian formalism. Note, that for combining all axes together, we write the kinetic energy as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{k}={\frac {1}{2}}\sum _{i}{\frac {|\mathbf {p} _{i}|^{2}}{2m_{i}}}=\sum _{i}\left({\frac {{p_{r}}_{i}^{2}}{2m_{i}}}+{\frac {1}{2}}{\mathbf {L} _{i}}^{\textsf {T}}{I_{i}}^{-1}\mathbf {L} _{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{k}={\frac {1}{2}}\sum _{i}{\frac {|\mathbf {p} _{i}|^{2}}{2m_{i}}}=\sum _{i}\left({\frac {{p_{r}}_{i}^{2}}{2m_{i}}}+{\frac {1}{2}}{\mathbf {L} _{i}}^{\textsf {T}}{I_{i}}^{-1}\mathbf {L} _{i}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b779494825d16a88de6fd83809bfa0e685cfbec9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.022ex; height:7.509ex;" alt="{\displaystyle E_{k}={\frac {1}{2}}\sum _{i}{\frac {|\mathbf {p} _{i}|^{2}}{2m_{i}}}=\sum _{i}\left({\frac {{p_{r}}_{i}^{2}}{2m_{i}}}+{\frac {1}{2}}{\mathbf {L} _{i}}^{\textsf {T}}{I_{i}}^{-1}\mathbf {L} _{i}\right)}"> where pr is the momentum in the radial direction, and the <a href="/wiki/Moment_of_inertia#Inertia_matrix_in_different_reference_frames" title="Moment of inertia">moment of inertia is a 3-dimensional matrix</a>; bold letters stand for 3-dimensional vectors. For point-like bodies we have: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{k}=\sum _{i}\left({\frac {{p_{r}}_{i}^{2}}{2m_{i}}}+{\frac {|{\mathbf {L} _{i}}|^{2}}{2m_{i}{r_{i}}^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{k}=\sum _{i}\left({\frac {{p_{r}}_{i}^{2}}{2m_{i}}}+{\frac {|{\mathbf {L} _{i}}|^{2}}{2m_{i}{r_{i}}^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7380e3e3a705ca0b46a1b795e69371969da9c357" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.746ex; height:7.509ex;" alt="{\displaystyle E_{k}=\sum _{i}\left({\frac {{p_{r}}_{i}^{2}}{2m_{i}}}+{\frac {|{\mathbf {L} _{i}}|^{2}}{2m_{i}{r_{i}}^{2}}}\right)}"> This form of the kinetic energy part of the Hamiltonian is useful in analyzing <a href="/wiki/Central_potential" class="mw-redirect" title="Central potential">central potential</a> problems, and is easily transformed to a <a href="/wiki/Quantum_mechanical" class="mw-redirect" title="Quantum mechanical">quantum mechanical</a> work frame (e.g. in the <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a> problem). <div class="mw-heading mw-heading2"><h2 id="Angular_momentum_in_orbital_mechanics">Angular momentum in orbital mechanics</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=14" title="Edit section: Angular momentum in orbital mechanics">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Specific_angular_momentum" title="Specific angular momentum">Specific angular momentum</a></div> While in classical mechanics the language of angular momentum can be replaced by Newton's laws of motion, it is particularly useful for motion in <a href="/wiki/Central_potential" class="mw-redirect" title="Central potential">central potential</a> such as planetary motion in the solar system. Thus, the orbit of a planet in the solar system is defined by its energy, angular momentum and angles of the orbit major axis relative to a coordinate frame. In astrodynamics and <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, a quantity closely related to angular momentum is defined as<a href="#cite_note-29">[29]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">h</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d21943fd68a6ac77006c0d7b71cedd7b3639b10" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.584ex; height:2.509ex;" alt="{\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} ,}"> called <a href="/wiki/Specific_angular_momentum" title="Specific angular momentum">specific angular momentum</a>. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =m\mathbf {h} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">h</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =m\mathbf {h} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451a7af5aa41456418306ab6657c780e020a4855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.879ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} =m\mathbf {h} .}"> <a href="/wiki/Mass" title="Mass">Mass</a> is often unimportant in orbital mechanics calculations, because motion of a body is determined by <a href="/wiki/Gravity" title="Gravity">gravity</a>. The primary body of the system is often so much larger than any bodies in motion about it that the gravitational effect of the smaller bodies on it can be neglected; it maintains, in effect, constant velocity. The motion of all bodies is affected by its gravity in the same way, regardless of mass, and therefore all move approximately the same way under the same conditions. <div class="mw-heading mw-heading2"><h2 id="Solid_bodies">Solid bodies</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=15" title="Edit section: Solid bodies">edit</a>]</div> Angular momentum is also an extremely useful concept for describing rotating rigid bodies such as a <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a> or a rocky planet. For a continuous mass distribution with <a href="/wiki/Density" title="Density">density</a> function ρ(r), a differential <a href="/wiki/Volume_element" title="Volume element">volume element</a> dV with <a href="/wiki/Position_vector" class="mw-redirect" title="Position vector">position vector</a> r within the mass has a mass element dm = ρ(r)dV. Therefore, the <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> angular momentum of this element is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {L} =\mathbf {r} \times dm\mathbf {v} =\mathbf {r} \times \rho (\mathbf {r} )dV\mathbf {v} =dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mi>d</mi> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>d</mi> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {L} =\mathbf {r} \times dm\mathbf {v} =\mathbf {r} \times \rho (\mathbf {r} )dV\mathbf {v} =dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/745bcb263750bb4b6df11130e916204d27d891f0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.669ex; height:2.843ex;" alt="{\displaystyle d\mathbf {L} =\mathbf {r} \times dm\mathbf {v} =\mathbf {r} \times \rho (\mathbf {r} )dV\mathbf {v} =dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} }"> and <a href="/wiki/Volume_integral" title="Volume integral">integrating</a> this <a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">differential</a> over the volume of the entire mass gives its total angular momentum: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\int _{V}dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>d</mi> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\int _{V}dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dfcf143a296d05052ee9f5612c8c78977806eea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.352ex; height:5.676ex;" alt="{\displaystyle \mathbf {L} =\int _{V}dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} }"> In the derivation which follows, integrals similar to this can replace the sums for the case of continuous mass. <div class="mw-heading mw-heading3"><h3 id="Collection_of_particles">Collection of particles</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=16" title="Edit section: Collection of particles">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ang_mom_vector_diagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Ang_mom_vector_diagram.png/220px-Ang_mom_vector_diagram.png" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Ang_mom_vector_diagram.png/330px-Ang_mom_vector_diagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Ang_mom_vector_diagram.png/440px-Ang_mom_vector_diagram.png 2x" data-file-width="698" data-file-height="682" /></a><figcaption>The angular momentum of the particles i is the sum of the cross products R × MV + Σri × mivi.</figcaption></figure> For a collection of particles in motion about an arbitrary origin, it is informative to develop the equation of angular momentum by resolving their motion into components about their own center of mass and about the origin. Given, <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec8e804f69706d3f5ad235f4f983220c8df7c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle m_{i}}"> is the mass of particle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f89b7e6f1eb0602ce2df26f016b4a4a9d55f86f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.803ex; height:2.509ex;" alt="{\displaystyle \mathbf {R} _{i}}"> is the position vector of particle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> w.r.t. the origin,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6961baf3efe47cd95e04a2e32f917e33fe787ef3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.819ex; height:2.509ex;" alt="{\displaystyle \mathbf {V} _{i}}"> is the velocity of particle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> w.r.t. the origin,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> is the position vector of the center of mass w.r.t. the origin,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0048514530d0c0fb8a7beb795110815a818784d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {V} }"> is the velocity of the center of mass w.r.t. the origin,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}"> is the position vector of particle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> w.r.t. the center of mass,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{i}}"> is the velocity of particle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> w.r.t. the center of mass,</li></ul> The total mass of the particles is simply their sum, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\sum _{i}m_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\sum _{i}m_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c71e78a6457b75b0af823f3989eabc616b624b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.77ex; height:5.509ex;" alt="{\displaystyle M=\sum _{i}m_{i}.}"> The position vector of the center of mass is defined by,<a href="#cite_note-30">[30]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\mathbf {R} =\sum _{i}m_{i}\mathbf {R} _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\mathbf {R} =\sum _{i}m_{i}\mathbf {R} _{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f133d75bb6cbe0958d66acc95e5e186655942e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.576ex; height:5.509ex;" alt="{\displaystyle M\mathbf {R} =\sum _{i}m_{i}\mathbf {R} _{i}.}"> By inspection, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{i}=\mathbf {R} +\mathbf {r} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{i}=\mathbf {R} +\mathbf {r} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6002070fa6d4ad65b2b6623f4f57918222bf11ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.647ex; height:2.509ex;" alt="{\displaystyle \mathbf {R} _{i}=\mathbf {R} +\mathbf {r} _{i}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V} _{i}=\mathbf {V} +\mathbf {v} _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} _{i}=\mathbf {V} +\mathbf {v} _{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f91b76b7f70b2baa710308cf0f00879cda12a9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.635ex; height:2.509ex;" alt="{\displaystyle \mathbf {V} _{i}=\mathbf {V} +\mathbf {v} _{i}.}"></dd></dl> The total angular momentum of the collection of particles is the sum of the angular momentum of each particle, <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: var(--color-success,#14866d); color: inherit;text-align: center; display: table"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\sum _{i}\left(\mathbf {R} _{i}\times m_{i}\mathbf {V} _{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\sum _{i}\left(\mathbf {R} _{i}\times m_{i}\mathbf {V} _{i}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e786c7cd2d1a5cc159a531caf1c8d6ae66911a34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.561ex; height:5.509ex;" alt="{\displaystyle \mathbf {L} =\sum _{i}\left(\mathbf {R} _{i}\times m_{i}\mathbf {V} _{i}\right)}">     (1) </div> Expanding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f89b7e6f1eb0602ce2df26f016b4a4a9d55f86f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.803ex; height:2.509ex;" alt="{\displaystyle \mathbf {R} _{i}}">, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left[\left(\mathbf {R} +\mathbf {r} _{i}\right)\times m_{i}\mathbf {V} _{i}\right]\\&=\sum _{i}\left[\mathbf {R} \times m_{i}\mathbf {V} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} _{i}\right]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left[\left(\mathbf {R} +\mathbf {r} _{i}\right)\times m_{i}\mathbf {V} _{i}\right]\\&=\sum _{i}\left[\mathbf {R} \times m_{i}\mathbf {V} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} _{i}\right]\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77077ef30b49ac992b78c6a4c6446709bb1720a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:34.239ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left[\left(\mathbf {R} +\mathbf {r} _{i}\right)\times m_{i}\mathbf {V} _{i}\right]\\&=\sum _{i}\left[\mathbf {R} \times m_{i}\mathbf {V} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} _{i}\right]\end{aligned}}}"></dd></dl> Expanding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6961baf3efe47cd95e04a2e32f917e33fe787ef3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.819ex; height:2.509ex;" alt="{\displaystyle \mathbf {V} _{i}}">, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left[\mathbf {R} \times m_{i}\left(\mathbf {V} +\mathbf {v} _{i}\right)+\mathbf {r} _{i}\times m_{i}(\mathbf {V} +\mathbf {v} _{i})\right]\\&=\sum _{i}\left[\mathbf {R} \times m_{i}\mathbf {V} +\mathbf {R} \times m_{i}\mathbf {v} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} +\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\right]\\&=\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {R} \times m_{i}\mathbf {v} _{i}+\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left[\mathbf {R} \times m_{i}\left(\mathbf {V} +\mathbf {v} _{i}\right)+\mathbf {r} _{i}\times m_{i}(\mathbf {V} +\mathbf {v} _{i})\right]\\&=\sum _{i}\left[\mathbf {R} \times m_{i}\mathbf {V} +\mathbf {R} \times m_{i}\mathbf {v} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} +\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\right]\\&=\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {R} \times m_{i}\mathbf {v} _{i}+\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4784bd78e4066783f47bd72472bfa49a0887b0bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:67.94ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left[\mathbf {R} \times m_{i}\left(\mathbf {V} +\mathbf {v} _{i}\right)+\mathbf {r} _{i}\times m_{i}(\mathbf {V} +\mathbf {v} _{i})\right]\\&=\sum _{i}\left[\mathbf {R} \times m_{i}\mathbf {V} +\mathbf {R} \times m_{i}\mathbf {v} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} +\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\right]\\&=\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {R} \times m_{i}\mathbf {v} _{i}+\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\end{aligned}}}"></dd></dl> It can be shown that (see sidebar), <table class="toccolours" style="float:right; margin-left:0.5em; margin-right:0.5em; font-size:84%; background:white; color:black; width:30em; max-width:30%;" cellspacing="5"> <tbody><tr> <td style="text-align:center;"> Prove that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9534099a1221dce32185b2cd617fe516c15618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.919ex; height:5.509ex;" alt="{\displaystyle \sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {R} _{i}-\mathbf {R} \\m_{i}\mathbf {r} _{i}&=m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\\sum _{i}m_{i}\mathbf {r} _{i}&=\sum _{i}m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\&=\sum _{i}(m_{i}\mathbf {R} _{i}-m_{i}\mathbf {R} )\\&=\sum _{i}m_{i}\mathbf {R} _{i}-\sum _{i}m_{i}\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-\left(\sum _{i}m_{i}\right)\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-M\mathbf {R} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {R} _{i}-\mathbf {R} \\m_{i}\mathbf {r} _{i}&=m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\\sum _{i}m_{i}\mathbf {r} _{i}&=\sum _{i}m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\&=\sum _{i}(m_{i}\mathbf {R} _{i}-m_{i}\mathbf {R} )\\&=\sum _{i}m_{i}\mathbf {R} _{i}-\sum _{i}m_{i}\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-\left(\sum _{i}m_{i}\right)\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-M\mathbf {R} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc97849976c304bb00a4ad72d312b17ee8f00a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -17.343ex; margin-bottom: -0.329ex; width:37.213ex; height:36.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {R} _{i}-\mathbf {R} \\m_{i}\mathbf {r} _{i}&=m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\\sum _{i}m_{i}\mathbf {r} _{i}&=\sum _{i}m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\&=\sum _{i}(m_{i}\mathbf {R} _{i}-m_{i}\mathbf {R} )\\&=\sum _{i}m_{i}\mathbf {R} _{i}-\sum _{i}m_{i}\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-\left(\sum _{i}m_{i}\right)\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-M\mathbf {R} \end{aligned}}}"> which, by the definition of the center of mass, is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2118daab5078173da3eebca1673a98c25d213869" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.983ex; height:2.509ex;" alt="{\displaystyle \mathbf {0} ,}"> and similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}m_{i}\mathbf {v} _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}m_{i}\mathbf {v} _{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68076abbb71360a37d3356756e3082a902115992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.338ex; height:3.009ex;" alt="{\textstyle \sum _{i}m_{i}\mathbf {v} _{i}.}"> </td></tr></tbody></table> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9534099a1221dce32185b2cd617fe516c15618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.919ex; height:5.509ex;" alt="{\displaystyle \sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}m_{i}\mathbf {v} _{i}=\mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}m_{i}\mathbf {v} _{i}=\mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd78a68dda7e71ed85c4882689859209270d92a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.875ex; height:5.509ex;" alt="{\displaystyle \sum _{i}m_{i}\mathbf {v} _{i}=\mathbf {0} ,}"></dd></dl> therefore the second and third terms vanish, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adda39d405afd75d159a23c9a06a46a21bf8f221" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.174ex; height:5.509ex;" alt="{\displaystyle \mathbf {L} =\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}.}"></dd></dl> The first term can be rearranged, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\mathbf {R} \times m_{i}\mathbf {V} =\mathbf {R} \times \sum _{i}m_{i}\mathbf {V} =\mathbf {R} \times M\mathbf {V} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\mathbf {R} \times m_{i}\mathbf {V} =\mathbf {R} \times \sum _{i}m_{i}\mathbf {V} =\mathbf {R} \times M\mathbf {V} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4100b708f2401085dd24879939ba51656f69e8d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.04ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\mathbf {R} \times m_{i}\mathbf {V} =\mathbf {R} \times \sum _{i}m_{i}\mathbf {V} =\mathbf {R} \times M\mathbf {V} ,}"></dd></dl> and total angular momentum for the collection of particles is finally,<a href="#cite_note-31">[31]</a> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: var(--color-success,#14866d); color: inherit;text-align: center; display: table"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\mathbf {R} \times M\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\mathbf {R} \times M\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803425b3b9c646bb3ec7305708051bb20f9e5f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.388ex; height:5.509ex;" alt="{\displaystyle \mathbf {L} =\mathbf {R} \times M\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}}">     (2) </div> The first term is the angular momentum of the center of mass relative to the origin. Similar to <a href="#Single_particle">§ Single particle</a>, below, it is the angular momentum of one particle of mass M at the center of mass moving with velocity V. The second term is the angular momentum of the particles moving relative to the center of mass, similar to <a href="#Fixed_center_of_mass">§ Fixed center of mass</a>, below. The result is general—the motion of the particles is not restricted to rotation or revolution about the origin or center of mass. The particles need not be individual masses, but can be elements of a continuous distribution, such as a solid body. Rearranging equation (<a href="#math_2">2</a>) by vector identities, multiplying both terms by "one", and grouping appropriately, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=M(\mathbf {R} \times \mathbf {V} )+\sum _{i}\left[m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right],\\&={\frac {R^{2}}{R^{2}}}M\left(\mathbf {R} \times \mathbf {V} \right)+\sum _{i}\left[{\frac {r_{i}^{2}}{r_{i}^{2}}}m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right],\\&=R^{2}M\left({\frac {\mathbf {R} \times \mathbf {V} }{R^{2}}}\right)+\sum _{i}\left[r_{i}^{2}m_{i}\left({\frac {\mathbf {r} _{i}\times \mathbf {v} _{i}}{r_{i}^{2}}}\right)\right],\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>M</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {L} &=M(\mathbf {R} \times \mathbf {V} )+\sum _{i}\left[m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right],\\&={\frac {R^{2}}{R^{2}}}M\left(\mathbf {R} \times \mathbf {V} \right)+\sum _{i}\left[{\frac {r_{i}^{2}}{r_{i}^{2}}}m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right],\\&=R^{2}M\left({\frac {\mathbf {R} \times \mathbf {V} }{R^{2}}}\right)+\sum _{i}\left[r_{i}^{2}m_{i}\left({\frac {\mathbf {r} _{i}\times \mathbf {v} _{i}}{r_{i}^{2}}}\right)\right],\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5324c7fa7e5e94893f678fab046c2538416bc51" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.579ex; margin-bottom: -0.259ex; width:49.353ex; height:20.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {L} &=M(\mathbf {R} \times \mathbf {V} )+\sum _{i}\left[m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right],\\&={\frac {R^{2}}{R^{2}}}M\left(\mathbf {R} \times \mathbf {V} \right)+\sum _{i}\left[{\frac {r_{i}^{2}}{r_{i}^{2}}}m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right],\\&=R^{2}M\left({\frac {\mathbf {R} \times \mathbf {V} }{R^{2}}}\right)+\sum _{i}\left[r_{i}^{2}m_{i}\left({\frac {\mathbf {r} _{i}\times \mathbf {v} _{i}}{r_{i}^{2}}}\right)\right],\\\end{aligned}}}"> gives the total angular momentum of the system of particles in terms of <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> and <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb8af7a2f64af348e559652b6b1f0d2415ba444" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\omega }}}">, <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: var(--color-success,#14866d); color: inherit;text-align: center; display: table"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =I_{R}{\boldsymbol {\omega }}_{R}+\sum _{i}I_{i}{\boldsymbol {\omega }}_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =I_{R}{\boldsymbol {\omega }}_{R}+\sum _{i}I_{i}{\boldsymbol {\omega }}_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea05c528015fbb4230312edc2addddce047ecdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.878ex; height:5.509ex;" alt="{\displaystyle \mathbf {L} =I_{R}{\boldsymbol {\omega }}_{R}+\sum _{i}I_{i}{\boldsymbol {\omega }}_{i}.}">     (3) </div> <div class="mw-heading mw-heading4"><h4 id="Single_particle_case">Single particle case</h4>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=17" title="Edit section: Single particle case">edit</a>]</div> In the case of a single particle moving about the arbitrary origin, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {v} _{i}=\mathbf {0} ,\\\mathbf {r} &=\mathbf {R} ,\\\mathbf {v} &=\mathbf {V} ,\\m&=M,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>M</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {v} _{i}=\mathbf {0} ,\\\mathbf {r} &=\mathbf {R} ,\\\mathbf {v} &=\mathbf {V} ,\\m&=M,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4257d4b04ffe20195d98c78c9963ba27e3e0aa86" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:13.183ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {v} _{i}=\mathbf {0} ,\\\mathbf {r} &=\mathbf {R} ,\\\mathbf {v} &=\mathbf {V} ,\\m&=M,\end{aligned}}}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/944a683c88b75f2a6c1988793c7f82ea7fb0b95f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.617ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\mathbf {0} ,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}I_{i}{\boldsymbol {\omega }}_{i}=\mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}I_{i}{\boldsymbol {\omega }}_{i}=\mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba328add1d5c9eea7ae394ed9c8ec4bb65d1214" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.115ex; height:5.509ex;" alt="{\displaystyle \sum _{i}I_{i}{\boldsymbol {\omega }}_{i}=\mathbf {0} ,}"> and equations (<a href="#math_2">2</a>) and (<a href="#math_3">3</a>) for total angular momentum reduce to, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\mathbf {R} \times m\mathbf {V} =I_{R}{\boldsymbol {\omega }}_{R}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\mathbf {R} \times m\mathbf {V} =I_{R}{\boldsymbol {\omega }}_{R}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c808752c6f3802a2598be401f3a190232f4763" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.007ex; height:2.509ex;" alt="{\displaystyle \mathbf {L} =\mathbf {R} \times m\mathbf {V} =I_{R}{\boldsymbol {\omega }}_{R}.}"> <div class="mw-heading mw-heading4"><h4 id="Case_of_a_fixed_center_of_mass">Case of a fixed center of mass</h4>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=18" title="Edit section: Case of a fixed center of mass">edit</a>]</div> For the case of the center of mass fixed in space with respect to the origin, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V} =\mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} =\mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8beda55f95c66718b581190ccc13aae081d2fd5b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.101ex; height:2.509ex;" alt="{\displaystyle \mathbf {V} =\mathbf {0} ,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} \times M\mathbf {V} =\mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} \times M\mathbf {V} =\mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3333259064e5dfb948d851f5b896b346d43eff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.387ex; height:2.509ex;" alt="{\displaystyle \mathbf {R} \times M\mathbf {V} =\mathbf {0} ,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{R}{\boldsymbol {\omega }}_{R}=\mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{R}{\boldsymbol {\omega }}_{R}=\mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1e57764d122bf124ac896dda2e31001565982a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.733ex; height:2.509ex;" alt="{\displaystyle I_{R}{\boldsymbol {\omega }}_{R}=\mathbf {0} ,}"> and equations (<a href="#math_2">2</a>) and (<a href="#math_3">3</a>) for total angular momentum reduce to, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\sum _{i}I_{i}{\boldsymbol {\omega }}_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\sum _{i}I_{i}{\boldsymbol {\omega }}_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf008451f7642e1b64ec6c32d2f5a1b538c9ac01" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.02ex; height:5.509ex;" alt="{\displaystyle \mathbf {L} =\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\sum _{i}I_{i}{\boldsymbol {\omega }}_{i}.}"> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Angular_momentum_in_general_relativity">Angular momentum in general relativity</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=19" title="Edit section: Angular momentum in general relativity">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Angular_momentum_bivector_and_pseudovector.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Angular_momentum_bivector_and_pseudovector.svg/220px-Angular_momentum_bivector_and_pseudovector.svg.png" decoding="async" width="220" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Angular_momentum_bivector_and_pseudovector.svg/330px-Angular_momentum_bivector_and_pseudovector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Angular_momentum_bivector_and_pseudovector.svg/440px-Angular_momentum_bivector_and_pseudovector.svg.png 2x" data-file-width="350" data-file-height="239" /></a><figcaption>The 3-angular momentum as a <a href="/wiki/Bivector" title="Bivector">bivector</a> (plane element) and <a href="/wiki/Axial_vector" class="mw-redirect" title="Axial vector">axial vector</a>, of a particle of mass m with instantaneous 3-position x and 3-momentum p.</figcaption></figure><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Relativistic_angular_momentum" title="Relativistic angular momentum">Relativistic angular momentum</a></div> In modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum – see <a href="#Angular_momentum_in_quantum_mechanics">below</a>) is described using a different formalism, instead of a classical <a href="/wiki/Pseudovector" title="Pseudovector">pseudovector</a>. In this formalism, angular momentum is the <a href="/wiki/2-form" class="mw-redirect" title="2-form">2-form</a> <a href="/wiki/Noether_charge" class="mw-redirect" title="Noether charge">Noether charge</a> associated with rotational invariance. As a result, angular momentum is generally not conserved locally for general <a href="/wiki/Curved_space" title="Curved space">curved spacetimes</a>, unless they have rotational symmetry;<a href="#cite_note-Hawking-32">[32]</a> whereas globally the notion of angular momentum itself only makes sense if the spacetime is asymptotically flat.<a href="#cite_note-Misner-33">[33]</a> If the spacetime is only axially symmetric like for the <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr metric</a>, the total angular momentum is not conserved but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\phi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b374d416af2eff24ab4f701fc6549a10dc0e4783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.471ex; height:2.343ex;" alt="{\displaystyle p_{\phi }}"> is conserved which is related to the invariance of rotating around the symmetry-axis, where note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\phi }=g_{\mu \phi }p^{\phi }=mg_{\mu \phi }dX^{\mu }/d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ϕ</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msup> <mo>=</mo> <mi>m</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ϕ</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\phi }=g_{\mu \phi }p^{\phi }=mg_{\mu \phi }dX^{\mu }/d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aed1956bacf1a475ea184f906bdb1624653ae8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:27.73ex; height:3.343ex;" alt="{\displaystyle p_{\phi }=g_{\mu \phi }p^{\phi }=mg_{\mu \phi }dX^{\mu }/d\tau }"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mu \nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bf4140993a891f5782167dc8a0c236dc7667b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.204ex; height:2.343ex;" alt="{\displaystyle g_{\mu \nu }}"> is the <a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">metric</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\sqrt {|p_{\mu }p^{\mu }|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\sqrt {|p_{\mu }p^{\mu }|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e615963330fa3441e5322dd47103b9c314770d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.542ex; height:4.843ex;" alt="{\displaystyle m={\sqrt {|p_{\mu }p^{\mu }|}}}"> is the <a href="/wiki/Rest_mass" class="mw-redirect" title="Rest mass">rest mass</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dX^{\mu }/d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dX^{\mu }/d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e567d77bea2220c7a9d674dfb3a9670ed25948" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.016ex; height:2.843ex;" alt="{\displaystyle dX^{\mu }/d\tau }"> is the <a href="/wiki/Four-velocity" title="Four-velocity">four-velocity</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\mu }=(t,r,\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{\mu }=(t,r,\theta ,\phi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d87dfa4faef9e85803b21313cde12ac73f8acfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.594ex; height:2.843ex;" alt="{\displaystyle X^{\mu }=(t,r,\theta ,\phi )}"> is the four-position in spherical coordinates. In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\mathbf {r} \wedge \mathbf {p} \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\mathbf {r} \wedge \mathbf {p} \,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a728ebb909aecdf46e42d9d76e7ef344fa0bf0ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.911ex; height:2.509ex;" alt="{\displaystyle \mathbf {L} =\mathbf {r} \wedge \mathbf {p} \,,}"> in which the <a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">exterior product</a> (∧) replaces the <a href="/wiki/Cross_product" title="Cross product">cross product</a> (×) (these products have similar characteristics but are nonequivalent). This has the advantage of a clearer geometric interpretation as a plane element, defined using the vectors x and p, and the expression is true in any number of dimensions. In Cartesian coordinates: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=\left(xp_{y}-yp_{x}\right)\mathbf {e} _{x}\wedge \mathbf {e} _{y}+\left(yp_{z}-zp_{y}\right)\mathbf {e} _{y}\wedge \mathbf {e} _{z}+\left(zp_{x}-xp_{z}\right)\mathbf {e} _{z}\wedge \mathbf {e} _{x}\\&=L_{xy}\mathbf {e} _{x}\wedge \mathbf {e} _{y}+L_{yz}\mathbf {e} _{y}\wedge \mathbf {e} _{z}+L_{zx}\mathbf {e} _{z}\wedge \mathbf {e} _{x}\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <mi>y</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <mi>z</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <mi>x</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {L} &=\left(xp_{y}-yp_{x}\right)\mathbf {e} _{x}\wedge \mathbf {e} _{y}+\left(yp_{z}-zp_{y}\right)\mathbf {e} _{y}\wedge \mathbf {e} _{z}+\left(zp_{x}-xp_{z}\right)\mathbf {e} _{z}\wedge \mathbf {e} _{x}\\&=L_{xy}\mathbf {e} _{x}\wedge \mathbf {e} _{y}+L_{yz}\mathbf {e} _{y}\wedge \mathbf {e} _{z}+L_{zx}\mathbf {e} _{z}\wedge \mathbf {e} _{x}\,,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/771af9385db3ca15515c2538fa62846f3fe6194e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:68.405ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\left(xp_{y}-yp_{x}\right)\mathbf {e} _{x}\wedge \mathbf {e} _{y}+\left(yp_{z}-zp_{y}\right)\mathbf {e} _{y}\wedge \mathbf {e} _{z}+\left(zp_{x}-xp_{z}\right)\mathbf {e} _{z}\wedge \mathbf {e} _{x}\\&=L_{xy}\mathbf {e} _{x}\wedge \mathbf {e} _{y}+L_{yz}\mathbf {e} _{y}\wedge \mathbf {e} _{z}+L_{zx}\mathbf {e} _{z}\wedge \mathbf {e} _{x}\,,\end{aligned}}}"> or more compactly in index notation: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{ij}=x_{i}p_{j}-x_{j}p_{i}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{ij}=x_{i}p_{j}-x_{j}p_{i}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32e4099346e97f23ae25a0c5424c77bfeaa02cce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.45ex; height:2.843ex;" alt="{\displaystyle L_{ij}=x_{i}p_{j}-x_{j}p_{i}\,.}"> The angular velocity can also be defined as an anti-symmetric second order tensor, with components ωij. The relation between the two anti-symmetric tensors is given by the moment of inertia which must now be a fourth order tensor:<a href="#cite_note-34">[34]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{ij}=I_{ijk\ell }\omega _{k\ell }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{ij}=I_{ijk\ell }\omega _{k\ell }\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c1df7926b52a248cb20e1277546a850aec2045" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.455ex; height:2.843ex;" alt="{\displaystyle L_{ij}=I_{ijk\ell }\omega _{k\ell }\,.}"> Again, this equation in L and ω as tensors is true in any number of dimensions. This equation also appears in the <a href="/wiki/Geometric_algebra" title="Geometric algebra">geometric algebra</a> formalism, in which L and ω are bivectors, and the moment of inertia is a mapping between them. In <a href="/wiki/Relativistic_mechanics" title="Relativistic mechanics">relativistic mechanics</a>, the <a href="/wiki/Relativistic_angular_momentum" title="Relativistic angular momentum">relativistic angular momentum</a> of a particle is expressed as an <a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">anti-symmetric tensor</a> of second order: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\alpha \beta }=X_{\alpha }P_{\beta }-X_{\beta }P_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\alpha \beta }=X_{\alpha }P_{\beta }-X_{\beta }P_{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0444666934b68983d172799a6719a3b12eb2f31" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.169ex; height:2.843ex;" alt="{\displaystyle M_{\alpha \beta }=X_{\alpha }P_{\beta }-X_{\beta }P_{\alpha }}"> in terms of <a href="/wiki/Four-vector" title="Four-vector">four-vectors</a>, namely the <a href="/wiki/Four-position" class="mw-redirect" title="Four-position">four-position</a> X and the <a href="/wiki/Four-momentum" title="Four-momentum">four-momentum</a> P, and absorbs the above L together with the <a href="/wiki/Moment_of_mass" class="mw-redirect" title="Moment of mass">moment of mass</a>, i.e., the product of the relativistic mass of the particle and its <a href="/wiki/Centre_of_mass" class="mw-redirect" title="Centre of mass">centre of mass</a>, which can be thought of as describing the motion of its centre of mass, since mass–energy is conserved. In each of the above cases, for a system of particles the total angular momentum is just the sum of the individual particle angular momenta, and the centre of mass is for the system. <div class="mw-heading mw-heading2"><h2 id="Angular_momentum_in_quantum_mechanics">Angular momentum in quantum mechanics</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=20" title="Edit section: Angular momentum in quantum mechanics">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">Angular momentum operator</a></div> In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, angular momentum (like other quantities) is expressed as an <a href="/wiki/Operator_(physics)" title="Operator (physics)">operator</a>, and its one-dimensional projections have <a href="/wiki/Point_spectrum" class="mw-redirect" title="Point spectrum">quantized eigenvalues</a>. Angular momentum is subject to the <a href="/wiki/Heisenberg_uncertainty_principle" class="mw-redirect" title="Heisenberg uncertainty principle">Heisenberg uncertainty principle</a>, implying that at any time, only one <a href="/wiki/Vector_projection" title="Vector projection">projection</a> (also called "component") can be measured with definite precision; the other two then remain uncertain. Because of this, the axis of rotation of a quantum particle is undefined. Quantum particles do possess a type of non-orbital angular momentum called "spin", but this angular momentum does not correspond to a spinning motion.<a href="#cite_note-35">[35]</a> In <a href="/wiki/Relativistic_quantum_mechanics#Relativistic_quantum_angular_momentum" title="Relativistic quantum mechanics">relativistic quantum mechanics</a> the above relativistic definition becomes a tensorial operator. <div class="mw-heading mw-heading3"><h3 id="Spin,_orbital,_and_total_angular_momentum">Spin, orbital, and total angular momentum</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=21" title="Edit section: Spin, orbital, and total angular momentum">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Spin_(physics)" title="Spin (physics)">Spin (physics)</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Classical_angular_momentum.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Classical_angular_momentum.svg/280px-Classical_angular_momentum.svg.png" decoding="async" width="280" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Classical_angular_momentum.svg/420px-Classical_angular_momentum.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Classical_angular_momentum.svg/560px-Classical_angular_momentum.svg.png 2x" data-file-width="666" data-file-height="527" /></a><figcaption>Angular momenta of a classical object.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li>Left: "spin" angular momentum S is really orbital angular momentum of the object at every point.</li><li>Right: extrinsic orbital angular momentum L about an axis.</li><li>Top: the <a href="/wiki/Moment_of_inertia_tensor" class="mw-redirect" title="Moment of inertia tensor">moment of inertia tensor</a> I and <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> ω (L is not always parallel to ω).<a href="#cite_note-36">[36]</a></li><li>Bottom: momentum p and its radial position r from the axis. The total angular momentum (spin plus orbital) is J. For a quantum particle the interpretations are different; <a href="/wiki/Spin_(physics)" title="Spin (physics)">particle spin</a> does not have the above interpretation.</li></ul></div> </figcaption></figure> The classical definition of angular momentum as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22179a9b81408e19de312b5fbfec30ff62cefa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.135ex; height:2.509ex;" alt="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }"> can be carried over to quantum mechanics, by reinterpreting r as the quantum <a href="/wiki/Position_operator" title="Position operator">position operator</a> and p as the quantum <a href="/wiki/Momentum_operator" title="Momentum operator">momentum operator</a>. L is then an <a href="/wiki/Operator_(physics)" title="Operator (physics)">operator</a>, specifically called the <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">orbital angular momentum operator</a>. The components of the angular momentum operator satisfy the commutation relations of the Lie algebra so(3). Indeed, these operators are precisely the infinitesimal action of the rotation group on the quantum <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>.<a href="#cite_note-37">[37]</a> (See also the discussion below of the angular momentum operators as the generators of rotations.) However, in quantum physics, there is another type of angular momentum, called spin angular momentum, represented by the spin operator S. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: spin is an intrinsic property of a particle, unrelated to any sort of motion in space and fundamentally different from orbital angular momentum. All <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a> have a characteristic spin (possibly zero),<a href="#cite_note-38">[38]</a> and almost all <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a> have nonzero spin.<a href="#cite_note-39">[39]</a> For example <a href="/wiki/Electron" title="Electron">electrons</a> have "spin 1/2" (this actually means "spin <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">ħ</a>/2"), <a href="/wiki/Photon" title="Photon">photons</a> have "spin 1" (this actually means "spin ħ"), and <a href="/wiki/Pion" title="Pion">pi-mesons</a> have spin 0.<a href="#cite_note-40">[40]</a> Finally, there is <a href="/wiki/Total_angular_momentum" class="mw-redirect" title="Total angular momentum">total angular momentum</a> J, which combines both the spin and orbital angular momentum of all particles and fields. (For one particle, J = L + S.) <a href="/wiki/Conservation_of_angular_momentum" class="mw-redirect" title="Conservation of angular momentum">Conservation of angular momentum</a> applies to J, but not to L or S; for example, the <a href="/wiki/Spin%E2%80%93orbit_interaction" title="Spin–orbit interaction">spin–orbit interaction</a> allows angular momentum to transfer back and forth between L and S, with the total remaining constant. Electrons and photons need not have integer-based values for total angular momentum, but can also have half-integer values.<a href="#cite_note-41">[41]</a> In molecules the total angular momentum F is the sum of the rovibronic (orbital) angular momentum N, the electron spin angular momentum S, and the nuclear spin angular momentum I. For electronic singlet states the rovibronic angular momentum is denoted J rather than N. As explained by Van Vleck,<a href="#cite_note-42">[42]</a> the components of the molecular rovibronic angular momentum referred to molecule-fixed axes have different commutation relations from those for the components about space-fixed axes. <div class="mw-heading mw-heading3"><h3 id="Quantization">Quantization</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=22" title="Edit section: Quantization">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">Angular momentum operator</a></div> In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, angular momentum is <a href="/wiki/Angular_momentum_quantization" class="mw-redirect" title="Angular momentum quantization">quantized</a> – that is, it cannot vary continuously, but only in "<a href="/wiki/Quantum_number" title="Quantum number">quantum leaps</a>" between certain allowed values. For any system, the following restrictions on measurement results apply, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }"> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d125dccc556f5c8b0bf98a4f3847590b3f353bd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}}"> is any <a href="/wiki/Euclidean_vector" title="Euclidean vector">Euclidean vector</a> such as x, y, or z: <table class="wikitable"> <tbody><tr> <td>If you <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">measure</a>... </td> <td>The result can be... </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\hat {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\hat {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b78f76128624fb10cd23012eb58a31e26541a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.801ex; height:2.843ex;" alt="{\displaystyle L_{\hat {n}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots ,-2\hbar ,-\hbar ,0,\hbar ,2\hbar ,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mo>−</mo> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mn>2</mn> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots ,-2\hbar ,-\hbar ,0,\hbar ,2\hbar ,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b19a282a732db5c4842105f212d6b5cfa0ba8b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.367ex; height:2.509ex;" alt="{\displaystyle \ldots ,-2\hbar ,-\hbar ,0,\hbar ,2\hbar ,\ldots }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\hat {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\hat {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79b82b532961ce7615933feec2b306ecad9ba832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.643ex; height:2.843ex;" alt="{\displaystyle S_{\hat {n}}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\hat {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{\hat {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17326aeae4fa4c823e540d305e80cc0f7f708625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.509ex; height:2.843ex;" alt="{\displaystyle J_{\hat {n}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots ,-{\frac {3}{2}}\hbar ,-\hbar ,-{\frac {1}{2}}\hbar ,0,{\frac {1}{2}}\hbar ,\hbar ,{\frac {3}{2}}\hbar ,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mo>−</mo> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots ,-{\frac {3}{2}}\hbar ,-\hbar ,-{\frac {1}{2}}\hbar ,0,{\frac {1}{2}}\hbar ,\hbar ,{\frac {3}{2}}\hbar ,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82abc8246e797f65c1206e7133672cc2a800f8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.525ex; height:5.176ex;" alt="{\displaystyle \ldots ,-{\frac {3}{2}}\hbar ,-\hbar ,-{\frac {1}{2}}\hbar ,0,{\frac {1}{2}}\hbar ,\hbar ,{\frac {3}{2}}\hbar ,\ldots }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&L^{2}\\={}&L_{x}^{2}+L_{y}^{2}+L_{z}^{2}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&L^{2}\\={}&L_{x}^{2}+L_{y}^{2}+L_{z}^{2}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1142965311188c2ec11536bcd0a200235f145a96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.915ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&L^{2}\\={}&L_{x}^{2}+L_{y}^{2}+L_{z}^{2}\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\hbar ^{2}n(n+1)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[\hbar ^{2}n(n+1)\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd720bdca13e8588818fb477347fb90c9789bb4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.917ex; height:3.343ex;" alt="{\displaystyle \left[\hbar ^{2}n(n+1)\right]}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0,1,2,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0,1,2,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a19cb2cfd4f9ebdbc8e5cbb9b92ecb9ace85cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;" alt="{\displaystyle n=0,1,2,\ldots }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6401d5d0155afb1406770d1eb80badce4e08ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{2}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fa18f680f853f4b3462401d4012c9b430b88b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.58ex; height:2.676ex;" alt="{\displaystyle J^{2}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\hbar ^{2}n(n+1)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[\hbar ^{2}n(n+1)\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd720bdca13e8588818fb477347fb90c9789bb4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.917ex; height:3.343ex;" alt="{\displaystyle \left[\hbar ^{2}n(n+1)\right]}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c261b8061ef8b0b1cfcf84fb9ab65b4c834704f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.993ex; height:3.509ex;" alt="{\displaystyle n=0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots }"> </td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circular_Standing_Wave.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Circular_Standing_Wave.gif/220px-Circular_Standing_Wave.gif" decoding="async" width="220" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/cf/Circular_Standing_Wave.gif 1.5x" data-file-width="295" data-file-height="279" /></a><figcaption>In this <a href="/wiki/Standing_wave" title="Standing wave">standing wave</a> on a circular string, the circle is broken into exactly 8 <a href="/wiki/Wavelength" title="Wavelength">wavelengths</a>. A standing wave like this can have 0,1,2, or any integer number of wavelengths around the circle, but it cannot have a non-integer number of wavelengths like 8.3. In quantum mechanics, angular momentum is quantized for a similar reason.</figcaption></figure> The <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }"> is tiny by everyday standards, about 10−34 <a href="/wiki/Joule-second" title="Joule-second">J s</a>, and therefore this quantization does not noticeably affect the angular momentum of macroscopic objects. However, it is very important in the microscopic world. For example, the structure of <a href="/wiki/Electron_shell" title="Electron shell">electron shells</a> and subshells in chemistry is significantly affected by the quantization of angular momentum. Quantization of angular momentum was first postulated by <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> in <a href="/wiki/Bohr_model" title="Bohr model">his model</a> of the atom and was later predicted by <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> in his <a href="/wiki/Schr%C3%B6dinger_equation#Quantization" title="Schrödinger equation">Schrödinger equation</a>. <div class="mw-heading mw-heading3"><h3 id="Uncertainty">Uncertainty</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=23" title="Edit section: Uncertainty">edit</a>]</div> In the definition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22179a9b81408e19de312b5fbfec30ff62cefa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.135ex; height:2.509ex;" alt="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }">, six operators are involved: The <a href="/wiki/Position_operator" title="Position operator">position operators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3705946c668b0bd4b21e7c19b95bb6ed05ea4ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.221ex; height:2.009ex;" alt="{\displaystyle r_{x}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e52760272851bdfd60c01877ee11bfc1857a60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.098ex; height:2.343ex;" alt="{\displaystyle r_{y}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07006192648caeccd5ad603f234b1f11e2e40b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.05ex; height:2.009ex;" alt="{\displaystyle r_{z}}">, and the <a href="/wiki/Momentum_operator" title="Momentum operator">momentum operators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5055ed65713825b48aa6ee05118c072e6f026a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.431ex; height:2.009ex;" alt="{\displaystyle p_{x}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1108b3eff778ca6f026e3e28c26cb093174cc2d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.308ex; height:2.343ex;" alt="{\displaystyle p_{y}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbd3c1a6173a7974e0095301da94447c5f67657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.261ex; height:2.009ex;" alt="{\displaystyle p_{z}}">. However, the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Heisenberg uncertainty principle</a> tells us that it is not possible for all six of these quantities to be known simultaneously with arbitrary precision. Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's <a href="/wiki/Magnitude_(vector)" class="mw-redirect" title="Magnitude (vector)">magnitude</a> and its component along one axis. The uncertainty is closely related to the fact that different components of an angular momentum operator do not <a href="/wiki/Commutator" title="Commutator">commute</a>, for example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{x}L_{y}\neq L_{y}L_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{x}L_{y}\neq L_{y}L_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14b5e35b0c4b63f2a2bd24c39f08e6e1c86f9528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.873ex; height:2.843ex;" alt="{\displaystyle L_{x}L_{y}\neq L_{y}L_{x}}">. (For the precise <a href="/wiki/Commutation_relation" class="mw-redirect" title="Commutation relation">commutation relations</a>, see <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">angular momentum operator</a>.) <div class="mw-heading mw-heading3"><h3 id="Total_angular_momentum_as_generator_of_rotations">Total angular momentum as generator of rotations</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=24" title="Edit section: Total angular momentum as generator of rotations">edit</a>]</div> As mentioned above, orbital angular momentum L is defined as in classical mechanics: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22179a9b81408e19de312b5fbfec30ff62cefa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.135ex; height:2.509ex;" alt="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }">, but total angular momentum J is defined in a different, more basic way: J is defined as the "generator of rotations".<a href="#cite_note-littlejohn-43">[43]</a> More specifically, J is defined so that the operator <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R({\hat {n}},\phi )\equiv \exp \left(-{\frac {i}{\hbar }}\phi \,\mathbf {J} \cdot {\hat {\mathbf {n} }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mi>ϕ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R({\hat {n}},\phi )\equiv \exp \left(-{\frac {i}{\hbar }}\phi \,\mathbf {J} \cdot {\hat {\mathbf {n} }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05d8a19780700ce3d93b241a525c4441f6479e34" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.728ex; height:6.176ex;" alt="{\displaystyle R({\hat {n}},\phi )\equiv \exp \left(-{\frac {i}{\hbar }}\phi \,\mathbf {J} \cdot {\hat {\mathbf {n} }}\right)}"> is the <a href="/wiki/Rotation_operator_(quantum_mechanics)" title="Rotation operator (quantum mechanics)">rotation operator</a> that takes any system and rotates it by angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> about the axis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {n} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae87b164ba005e99b51066c46d1eacc7f56564a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}">. (The "exp" in the formula refers to <a href="/wiki/Matrix_exponential" title="Matrix exponential">operator exponential</a>.) To put this the other way around, whatever our quantum Hilbert space is, we expect that the <a href="/wiki/Rotation_group_SO(3)" class="mw-redirect" title="Rotation group SO(3)">rotation group SO(3)</a> will act on it. There is then an associated action of the Lie algebra so(3) of SO(3); the operators describing the action of so(3) on our Hilbert space are the (total) angular momentum operators. The relationship between the angular momentum operator and the rotation operators is the same as the relationship between <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> in mathematics. The close relationship between angular momentum and rotations is reflected in <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a> that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant. <div class="mw-heading mw-heading2"><h2 id="Angular_momentum_in_electrodynamics">Angular momentum in electrodynamics</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=25" title="Edit section: Angular momentum in electrodynamics">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Momentum#Particle_in_a_field" title="Momentum">Momentum (Particle in field)</a></div> When describing the motion of a <a href="/wiki/Charged_particle" title="Charged particle">charged particle</a> in an <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a>, the <a href="/wiki/Canonical_momentum" class="mw-redirect" title="Canonical momentum">canonical momentum</a> P (derived from the <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a> for this system) is not <a href="/wiki/Gauge_invariant" class="mw-redirect" title="Gauge invariant">gauge invariant</a>. As a consequence, the canonical angular momentum L = r × P is not gauge invariant either. Instead, the momentum that is physical, the so-called kinetic momentum (used throughout this article), is (in <a href="/wiki/SI_units" class="mw-redirect" title="SI units">SI units</a>) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =m\mathbf {v} =\mathbf {P} -e\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>−</mo> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =m\mathbf {v} =\mathbf {P} -e\mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b13cdc37559fc7edc45f19ca88d0763bb4e5000" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.904ex; height:2.509ex;" alt="{\displaystyle \mathbf {p} =m\mathbf {v} =\mathbf {P} -e\mathbf {A} }"> where e is the <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> of the particle and A the <a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">magnetic vector potential</a> of the electromagnetic field. The gauge-invariant angular momentum, that is kinetic angular momentum, is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {K} =\mathbf {r} \times (\mathbf {P} -e\mathbf {A} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>−</mo> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {K} =\mathbf {r} \times (\mathbf {P} -e\mathbf {A} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ea5e7e0c75ea3090123ba971034a6cd72db6b79" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.714ex; height:2.843ex;" alt="{\displaystyle \mathbf {K} =\mathbf {r} \times (\mathbf {P} -e\mathbf {A} )}"> The interplay with quantum mechanics is discussed further in the article on <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relations</a>. <div class="mw-heading mw-heading2"><h2 id="Angular_momentum_in_optics">Angular momentum in optics</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=26" title="Edit section: Angular momentum in optics">edit</a>]</div> In classical Maxwell electrodynamics the <a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a> is a linear momentum density of electromagnetic field.<a href="#cite_note-Okulov2008-44">[44]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {S} (\mathbf {r} ,t)=\epsilon _{0}c^{2}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} (\mathbf {r} ,t)=\epsilon _{0}c^{2}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1def5c88bdb99e90ec83e577815c9a68fdf6b4e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.143ex; height:3.176ex;" alt="{\displaystyle \mathbf {S} (\mathbf {r} ,t)=\epsilon _{0}c^{2}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t).}"> The angular momentum density vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} (\mathbf {r} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} (\mathbf {r} ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dac2bad85a420ba827c3449d04bfd020224dacad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.393ex; height:2.843ex;" alt="{\displaystyle \mathbf {L} (\mathbf {r} ,t)}"> is given by a vector product as in classical mechanics:<a href="#cite_note-Okulov2008J-45">[45]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} (\mathbf {r} ,t)=\epsilon _{0}\mu _{0}\mathbf {r} \times \mathbf {S} (\mathbf {r} ,t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} (\mathbf {r} ,t)=\epsilon _{0}\mu _{0}\mathbf {r} \times \mathbf {S} (\mathbf {r} ,t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32730883e04c59aa86ae4bd2c0ed75987fec4b1f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.806ex; height:2.843ex;" alt="{\displaystyle \mathbf {L} (\mathbf {r} ,t)=\epsilon _{0}\mu _{0}\mathbf {r} \times \mathbf {S} (\mathbf {r} ,t).}"> The above identities are valid locally, i.e. in each space point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> in a given moment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">. <div class="mw-heading mw-heading2"><h2 id="Angular_momentum_in_nature_and_the_cosmos">Angular momentum in nature and the cosmos</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=27" title="Edit section: Angular momentum in nature and the cosmos">edit</a>]</div> <a href="/wiki/Tropical_cyclones" class="mw-redirect" title="Tropical cyclones">Tropical cyclones</a> and other related weather phenomena involve conservation of angular momentum in order to explain the dynamics. Winds revolve slowly around low pressure systems, mainly due to the <a href="/wiki/Coriolis_force#Meteorology" title="Coriolis force">coriolis</a> effect. If the low pressure intensifies and the slowly circulating air is drawn toward the center, the molecules must speed up in order to conserve angular momentum. By the time they reach the center, the speeds become destructive.<a href="#cite_note-Tropical_Cyclone_Structure-2">[2]</a> <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> determined the laws of planetary motion without knowledge of conservation of momentum. However, not long after his discovery their derivation was determined from conservation of angular momentum. Planets move more slowly the further they are out in their elliptical orbits, which is explained intuitively by the fact that orbital angular momentum is proportional to the radius of the orbit. Since the mass does not change and the angular momentum is conserved, the velocity drops. <a href="/wiki/Tidal_acceleration" title="Tidal acceleration">Tidal acceleration</a> is an effect of the tidal forces between an orbiting natural satellite (e.g. the <a href="/wiki/Moon" title="Moon">Moon</a>) and the primary planet that it orbits (e.g. Earth). The gravitational torque between the Moon and the tidal bulge of Earth causes the Moon to be constantly promoted to a slightly higher orbit (~3.8 cm per year) and Earth to be <a href="/wiki/%CE%94T_(timekeeping)#Values_prior_to_1955" title="ΔT (timekeeping)">decelerated</a> (by −25.858 ± 0.003″/cy²) in its rotation (the <a href="/wiki/%CE%94T_(timekeeping)#Universal_time" title="ΔT (timekeeping)">length of the day increases</a> by ~1.7 ms per century, +2.3 ms from tidal effect and −0.6 ms from post-glacial rebound). The Earth loses angular momentum which is transferred to the Moon such that the overall angular momentum is conserved. <div class="mw-heading mw-heading2"><h2 id="Angular_momentum_in_engineering_and_technology">Angular momentum in engineering and technology</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=28" title="Edit section: Angular momentum in engineering and technology">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm/220px--Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="105" data-mwtitle="Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bf/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bf/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/b/bf/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm" type="video/webm; codecs="vp8, vorbis"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bf/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bf/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bf/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bf/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm/Video_of_a_complete_use_session_with_a_gyroscopic_exercise_tool.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video><figcaption>Video: A <a href="/wiki/Gyroscopic_exercise_tool" title="Gyroscopic exercise tool">gyroscopic exercise tool</a> is an application of the conservation of angular momentum for muscle strengthening. A mass quickly rotating about its axis in a ball-shaped device defines an angular momentum. When the person exercising tilts the ball, a force results which even increases the rotational speed when reacted to specifically by the user.</figcaption></figure> Examples of using conservation of angular momentum for practical advantage are abundant. In engines such as <a href="/wiki/Steam_engines" class="mw-redirect" title="Steam engines">steam engines</a> or <a href="/wiki/Internal_combustion_engines" class="mw-redirect" title="Internal combustion engines">internal combustion engines</a>, a <a href="/wiki/Flywheel" title="Flywheel">flywheel</a> is needed to efficiently convert the lateral motion of the pistons to rotational motion. <a href="/wiki/Inertial_navigation_system" title="Inertial navigation system">Inertial navigation systems</a> explicitly use the fact that angular momentum is conserved with respect to the <a href="/wiki/Inertial_frame" class="mw-redirect" title="Inertial frame">inertial frame</a> of space. Inertial navigation is what enables submarine trips under the polar ice cap, but are also crucial to all forms of modern navigation. <a href="/wiki/Rifle" title="Rifle">Rifled bullets</a> use the stability provided by conservation of angular momentum to be more true in their trajectory. The invention of rifled firearms and cannons gave their users significant strategic advantage in battle, and thus were a technological turning point in history. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=29" title="Edit section: History">edit</a>]</div> <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>, in the <a href="/wiki/Philosophiae_Naturalis_Principia_Mathematica" class="mw-redirect" title="Philosophiae Naturalis Principia Mathematica">Principia</a>, hinted at angular momentum in his examples of the <a href="/wiki/First_law_of_motion" class="mw-redirect" title="First law of motion">first law of motion</a>,<blockquote>A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.<a href="#cite_note-46">[46]</a></blockquote>He did not further investigate angular momentum directly in the Principia, saying:<blockquote>From such kind of reflexions also sometimes arise the circular motions of bodies about their own centres. But these are cases which I do not consider in what follows; and it would be too tedious to demonstrate every particular that relates to this subject.<a href="#cite_note-47">[47]</a></blockquote>However, his geometric proof of the <a href="/wiki/Kepler%27s_laws_of_planetary_motion#Second_law" title="Kepler's laws of planetary motion">law of areas</a> is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a <a href="/wiki/Central_force" title="Central force">central force</a>. <div class="mw-heading mw-heading3"><h3 id="The_Law_of_Areas">The Law of Areas</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=30" title="Edit section: The Law of Areas">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Classical_central-force_problem" title="Classical central-force problem">Classical central-force problem</a> and <a href="/wiki/Areal_velocity" title="Areal velocity">Areal velocity</a></div> <div class="mw-heading mw-heading4"><h4 id="Newton's_derivation">Newton's derivation</h4>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=31" title="Edit section: Newton's derivation">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Newton_area_law_derivation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/73/Newton_area_law_derivation.gif" decoding="async" width="256" height="256" class="mw-file-element" data-file-width="256" data-file-height="256" /></a><figcaption>Newton's derivation of the area law using geometric means</figcaption></figure> As a <a href="/wiki/Planet" title="Planet">planet</a> orbits the <a href="/wiki/Sun" title="Sun">Sun</a>, the line between the Sun and the planet sweeps out equal areas in equal intervals of time. This had been known since Kepler expounded his <a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">second law of planetary motion</a>. Newton derived a unique geometric proof, and went on to show that the attractive force of the Sun's <a href="/wiki/Gravity" title="Gravity">gravity</a> was the cause of all of Kepler's laws. During the first interval of time, an object is in motion from point A to point B. Undisturbed, it would continue to point c during the second interval. When the object arrives at B, it receives an impulse directed toward point S. The impulse gives it a small added velocity toward S, such that if this were its only velocity, it would move from B to V during the second interval. By the <a href="/wiki/Parallelogram_of_force" title="Parallelogram of force">rules of velocity composition</a>, these two velocities add, and point C is found by construction of parallelogram BcCV. Thus the object's path is deflected by the impulse so that it arrives at point C at the end of the second interval. Because the triangles SBc and SBC have the same base SB and the same height Bc or VC, they have the same area. By symmetry, triangle SBc also has the same area as triangle SAB, therefore the object has swept out equal areas SAB and SBC in equal times. At point C, the object receives another impulse toward S, again deflecting its path during the third interval from d to D. Thus it continues to E and beyond, the triangles SAB, SBc, SBC, SCd, SCD, SDe, SDE all having the same area. Allowing the time intervals to become ever smaller, the path ABCDE approaches indefinitely close to a continuous curve. Note that because this derivation is geometric, and no specific force is applied, it proves a more general law than Kepler's second law of planetary motion. It shows that the Law of Areas applies to any central force, attractive or repulsive, continuous or non-continuous, or zero. <div class="mw-heading mw-heading4"><h4 id="Conservation_of_angular_momentum_in_the_Law_of_Areas">Conservation of angular momentum in the Law of Areas</h4>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=32" title="Edit section: Conservation of angular momentum in the Law of Areas">edit</a>]</div> The proportionality of angular momentum to the area swept out by a moving object can be understood by realizing that the bases of the triangles, that is, the lines from S to the object, are equivalent to the <a href="#Scalar_–_angular_momentum_in_two_dimensions">radius <var>r</var></a>, and that the heights of the triangles are proportional to the perpendicular component of <a href="#Scalar_–_angular_momentum_in_two_dimensions">velocity <var>v</var>⊥</a>. Hence, if the area swept per unit time is constant, then by the triangular area formula <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/2⁠(base)(height), the product (base)(height) and therefore the product <var>rv</var>⊥ are constant: if <var>r</var> and the base length are decreased, <var>v</var>⊥ and height must increase proportionally. Mass is constant, therefore <a href="#Scalar_–_angular_momentum_in_two_dimensions">angular momentum <var>rmv</var>⊥</a> is conserved by this exchange of distance and velocity. In the case of triangle SBC, area is equal to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠(SB)(VC). Wherever C is eventually located due to the impulse applied at B, the product (SB)(VC), and therefore <var>rmv</var>⊥ remain constant. Similarly so for each of the triangles. Another areal proof of conservation of angular momentum for any central force uses Mamikon's sweeping tangents theorem.<a href="#cite_note-48">[48]</a><a href="#cite_note-49">[49]</a> <div class="mw-heading mw-heading3"><h3 id="After_Newton">After Newton</h3>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=33" title="Edit section: After Newton">edit</a>]</div> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a>, and <a href="/wiki/Patrick_d%27Arcy" title="Patrick d'Arcy">Patrick d'Arcy</a> all understood angular momentum in terms of conservation of <a href="/wiki/Areal_velocity" title="Areal velocity">areal velocity</a>, a result of their analysis of Kepler's second law of planetary motion. It is unlikely that they realized the implications for ordinary rotating matter.<a href="#cite_note-50">[50]</a> In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his <a href="/wiki/Mechanica" title="Mechanica">Mechanica</a> without further developing them.<a href="#cite_note-51">[51]</a> Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.<a href="#cite_note-52">[52]</a> In 1799, <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> first realized that a fixed plane was associated with rotation—his <a href="/wiki/Invariable_plane" title="Invariable plane">invariable plane</a>. <a href="/wiki/Louis_Poinsot" title="Louis Poinsot">Louis Poinsot</a> in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments". In 1852 <a href="/wiki/L%C3%A9on_Foucault" title="Léon Foucault">Léon Foucault</a> used a <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a> in an experiment to display the Earth's rotation. <a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">William J. M. Rankine's</a> 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:<blockquote>...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation.</blockquote>In an 1872 edition of the same book, Rankine stated that "The term angular momentum was introduced by Mr. Hayward,"<a href="#cite_note-53">[53]</a> probably referring to R.B. Hayward's article On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space with Applications,<a href="#cite_note-Hayward-54">[54]</a> which was introduced in 1856, and published in 1864. Rankine was mistaken, as numerous publications feature the term starting in the late 18th to early 19th centuries.<a href="#cite_note-55">[55]</a> However, Hayward's article apparently was the first use of the term and the concept seen by much of the English-speaking world. Before this, angular momentum was typically referred to as "momentum of rotation" in English.<a href="#cite_note-56">[56]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=34" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/Absolute_angular_momentum" title="Absolute angular momentum">Absolute angular momentum</a></li> <li><a href="/wiki/Angular_momentum_coupling" title="Angular momentum coupling">Angular momentum coupling</a></li> <li><a href="/wiki/Angular_momentum_of_light" title="Angular momentum of light">Angular momentum of light</a></li> <li><a href="/wiki/Angular_momentum_diagrams_(quantum_mechanics)" title="Angular momentum diagrams (quantum mechanics)">Angular momentum diagrams (quantum mechanics)</a></li> <li><a href="/wiki/Chaotic_rotation" title="Chaotic rotation">Chaotic rotation</a></li> <li><a href="/wiki/Clebsch%E2%80%93Gordan_coefficients" title="Clebsch–Gordan coefficients">Clebsch–Gordan coefficients</a></li> <li><a href="/wiki/List_of_equations_in_classical_mechanics#Dynamics" title="List of equations in classical mechanics">Linear-rotational analogs</a></li> <li><a href="/wiki/Orders_of_magnitude_(angular_momentum)" title="Orders of magnitude (angular momentum)">Orders of magnitude (angular momentum)</a></li> <li><a href="/wiki/Pauli%E2%80%93Lubanski_pseudovector" title="Pauli–Lubanski pseudovector">Pauli–Lubanski pseudovector</a></li> <li><a href="/wiki/Relativistic_angular_momentum" title="Relativistic angular momentum">Relativistic angular momentum</a></li> <li><a href="/wiki/Rigid_rotor" title="Rigid rotor">Rigid rotor</a></li> <li><a href="/wiki/Rotational_energy" title="Rotational energy">Rotational energy</a></li> <li><a href="/wiki/Specific_relative_angular_momentum" class="mw-redirect" title="Specific relative angular momentum">Specific relative angular momentum</a></li> <li><a href="/wiki/Yrast" title="Yrast">Yrast</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=35" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a 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Scientific American. August 9, 2012. Retrieved January 4, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Soaring+Science%3A+The+Aerodynamics+of+Flying+a+Frisbee&rft.pub=Scientific+American&rft.date=2012-08-09&rft_id=https%3A%2F%2Fwww.scientificamerican.com%2Farticle%2Fbring-science-home-frisbee-aerodynamics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-Tropical_Cyclone_Structure-2">^ <a href="#cite_ref-Tropical_Cyclone_Structure_2-0">a</a> <a href="#cite_ref-Tropical_Cyclone_Structure_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.weather.gov/jetstream/tc_structure">"Tropical Cyclone Structure"</a>. National Weather Service. 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Philosophical Transactions of the Royal Society. 313 (1524): 47–70. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1984RSPTA.313...47S">1984RSPTA.313...47S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1984.0082">10.1098/rsta.1984.0082</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120566848">120566848</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society&rft.atitle=Long-term+changes+in+the+rotation+of+the+earth+%E2%80%93+700+B.C.+to+A.D.+1980&rft.volume=313&rft.issue=1524&rft.pages=47-70&rft.date=1984&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120566848%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frsta.1984.0082&rft_id=info%3Abibcode%2F1984RSPTA.313...47S&rft.aulast=Stephenson&rft.aufirst=F.+R.&rft.au=Morrison%2C+L.+V.&rft.au=Whitrow%2C+G.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> +2.40 ms/century divided by 36525 days. </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDickey,_J._O.1994" class="citation journal cs1">Dickey, J. O.; et al. (1994). <a rel="nofollow" class="external text" href="http://physics.ucsd.edu/~tmurphy/apollo/doc/Dickey.pdf">"Lunar Laser Ranging: A Continuing Legacy of the Apollo Program"</a> (PDF). Science. 265 (5171): 482–90, see 486. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994Sci...265..482D">1994Sci...265..482D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.265.5171.482">10.1126/science.265.5171.482</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17781305">17781305</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10157934">10157934</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://physics.ucsd.edu/~tmurphy/apollo/doc/Dickey.pdf">Archived</a> (PDF) from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Lunar+Laser+Ranging%3A+A+Continuing+Legacy+of+the+Apollo+Program&rft.volume=265&rft.issue=5171&rft.pages=482-90%2C+see+486&rft.date=1994&rft_id=info%3Adoi%2F10.1126%2Fscience.265.5171.482&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10157934%23id-name%3DS2CID&rft_id=info%3Apmid%2F17781305&rft_id=info%3Abibcode%2F1994Sci...265..482D&rft.au=Dickey%2C+J.+O.&rft_id=http%3A%2F%2Fphysics.ucsd.edu%2F~tmurphy%2Fapollo%2Fdoc%2FDickey.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz1995" class="citation book cs1">Landau, L. D.; Lifshitz, E. M. (1995). The classical theory of fields. Course of Theoretical Physics. Oxford, Butterworth–Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-2768-9" title="Special:BookSources/978-0-7506-2768-9"><bdi>978-0-7506-2768-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+classical+theory+of+fields&rft.series=Course+of+Theoretical+Physics&rft.pub=Oxford%2C+Butterworth%E2%80%93Heinemann&rft.date=1995&rft.isbn=978-0-7506-2768-9&rft.aulast=Landau&rft.aufirst=L.+D.&rft.au=Lifshitz%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> Tenenbaum, M., & Pollard, H. (1985). Ordinary differential equations en elementary textbook for students of mathematics. Engineering and the Sciences. </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamond2020" class="citation book cs1">Ramond, Pierre (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aXr-DwAAQBAJ">Field Theory: A Modern Primer</a> (2nd ed.). Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780429689017" title="Special:BookSources/9780429689017"><bdi>9780429689017</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Field+Theory%3A+A+Modern+Primer&rft.edition=2nd&rft.pub=Routledge&rft.date=2020&rft.isbn=9780429689017&rft.aulast=Ramond&rft.aufirst=Pierre&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaXr-DwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=aXr-DwAAQBAJ&pg=PA1">Extract of page 1</a> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Morin2008" class="citation book cs1">David Morin (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ni6CD7K2X4MC">Introduction to Classical Mechanics: With Problems and Solutions</a>. Cambridge University Press. p. 311. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-46837-4" title="Special:BookSources/978-1-139-46837-4"><bdi>978-1-139-46837-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Classical+Mechanics%3A+With+Problems+and+Solutions&rft.pages=311&rft.pub=Cambridge+University+Press&rft.date=2008&rft.isbn=978-1-139-46837-4&rft.au=David+Morin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNi6CD7K2X4MC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ni6CD7K2X4MC&pg=PA311">Extract of page 311</a> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBattin1999" class="citation book cs1">Battin, Richard H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition. American Institute of Aeronautics and Astronautics, Inc. p. 115. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56347-342-5" title="Special:BookSources/978-1-56347-342-5"><bdi>978-1-56347-342-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Mathematics+and+Methods+of+Astrodynamics%2C+Revised+Edition&rft.pages=115&rft.pub=American+Institute+of+Aeronautics+and+Astronautics%2C+Inc.&rft.date=1999&rft.isbn=978-1-56347-342-5&rft.aulast=Battin&rft.aufirst=Richard+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilson1915" class="citation journal cs1">Wilson, E. B. (1915). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nsI0AAAAIAAJ">"Linear Momentum, Kinetic Energy and Angular Momentum"</a>. The American Mathematical Monthly. XXII. Ginn and Co., Boston, in cooperation with University of Chicago, et al.: 188, equation (3).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Linear+Momentum%2C+Kinetic+Energy+and+Angular+Momentum&rft.volume=XXII&rft.pages=188%2C+equation+%283%29&rft.date=1915&rft.aulast=Wilson&rft.aufirst=E.+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnsI0AAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilson1915" class="citation journal cs1">Wilson, E. B. (1915). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nsI0AAAAIAAJ">"Linear Momentum, Kinetic Energy and Angular Momentum"</a>. The American Mathematical Monthly. XXII. Ginn and Co., Boston, in cooperation with University of Chicago, et al.: 191, Theorem 8.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Linear+Momentum%2C+Kinetic+Energy+and+Angular+Momentum&rft.volume=XXII&rft.pages=191%2C+Theorem+8&rft.date=1915&rft.aulast=Wilson&rft.aufirst=E.+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnsI0AAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-Hawking-32"><a href="#cite_ref-Hawking_32-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHawkingEllis1973" class="citation book cs1">Hawking, S. W.; Ellis, G. F. R. (1973). <a rel="nofollow" class="external text" href="https://doi.org/10.1017/CBO9780511524646">The Large Scale Structure of Space-Time</a>. Cambridge University Press. pp. 62–63. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511524646">10.1017/CBO9780511524646</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-09906-6" title="Special:BookSources/978-0-521-09906-6"><bdi>978-0-521-09906-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Large+Scale+Structure+of+Space-Time&rft.pages=62-63&rft.pub=Cambridge+University+Press&rft.date=1973&rft_id=info%3Adoi%2F10.1017%2FCBO9780511524646&rft.isbn=978-0-521-09906-6&rft.aulast=Hawking&rft.aufirst=S.+W.&rft.au=Ellis%2C+G.+F.+R.&rft_id=https%3A%2F%2Fdoi.org%2F10.1017%2FCBO9780511524646&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-Misner-33"><a href="#cite_ref-Misner_33-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMisnerThorneWheeler1973" class="citation book cs1">Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1973). "20: Conservation laws for 4-momentum and angular momentum". <a rel="nofollow" class="external text" href="https://archive.org/details/GravitationMisnerThorneWheeler">Gravitation</a>. W. H. Freeman and Company.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=20%3A+Conservation+laws+for+4-momentum+and+angular+momentum&rft.btitle=Gravitation&rft.pub=W.+H.+Freeman+and+Company&rft.date=1973&rft.aulast=Misner&rft.aufirst=C.+W.&rft.au=Thorne%2C+K.+S.&rft.au=Wheeler%2C+J.+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FGravitationMisnerThorneWheeler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> Synge and Schild, Tensor calculus, Dover publications, 1978 edition, p. 161. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-63612-2" title="Special:BookSources/978-0-486-63612-2">978-0-486-63612-2</a>. </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Podesta2002" class="citation book cs1">de Podesta, Michael (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ya-4enCDoWQC">Understanding the Properties of Matter</a> (2nd, illustrated, revised ed.). CRC Press. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ya-4enCDoWQC&pg=PA29">29</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-25788-6" title="Special:BookSources/978-0-415-25788-6"><bdi>978-0-415-25788-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+the+Properties+of+Matter&rft.pages=29&rft.edition=2nd%2C+illustrated%2C+revised&rft.pub=CRC+Press&rft.date=2002&rft.isbn=978-0-415-25788-6&rft.aulast=de+Podesta&rft.aufirst=Michael&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dya-4enCDoWQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR.P._FeynmanR.B._LeightonM._Sands1964" class="citation book cs1">R.P. Feynman; R.B. Leighton; M. Sands (1964). Feynman's Lectures on Physics (volume 2). Addison–Wesley. pp. 31–7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-02117-2" title="Special:BookSources/978-0-201-02117-2"><bdi>978-0-201-02117-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Feynman%27s+Lectures+on+Physics+%28volume+2%29&rft.pages=31-7&rft.pub=Addison%E2%80%93Wesley&rft.date=1964&rft.isbn=978-0-201-02117-2&rft.au=R.P.+Feynman&rft.au=R.B.+Leighton&rft.au=M.+Sands&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <a href="#CITEREFHall2013">Hall 2013</a> Section 17.3 </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVeltman2018" class="citation book cs1">Veltman, Martinus J G (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xWdhDwAAQBAJ&pg=PT351">Facts And Mysteries In Elementary Particle Physics</a> (revised ed.). World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-323-707-0" title="Special:BookSources/978-981-323-707-0"><bdi>978-981-323-707-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Facts+And+Mysteries+In+Elementary+Particle+Physics&rft.edition=revised&rft.pub=World+Scientific&rft.date=2018&rft.isbn=978-981-323-707-0&rft.aulast=Veltman&rft.aufirst=Martinus+J+G&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxWdhDwAAQBAJ%26pg%3DPT351&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThaller2005" class="citation book cs1">Thaller, Bernd (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iq1Gi6hmTRAC">Advanced Visual Quantum Mechanics</a> (illustrated ed.). Springer Science & Business Media. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iq1Gi6hmTRAC&pg=PA114">114</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-27127-9" title="Special:BookSources/978-0-387-27127-9"><bdi>978-0-387-27127-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Visual+Quantum+Mechanics&rft.pages=114&rft.edition=illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2005&rft.isbn=978-0-387-27127-9&rft.aulast=Thaller&rft.aufirst=Bernd&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Diq1Gi6hmTRAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrange1998" class="citation book cs1">Strange, Paul (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sdVrBM2w0Ow">Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics</a> (illustrated ed.). Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sdVrBM2w0OwC&pg=PA64">64</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-56583-7" title="Special:BookSources/978-0-521-56583-7"><bdi>978-0-521-56583-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativistic+Quantum+Mechanics%3A+With+Applications+in+Condensed+Matter+and+Atomic+Physics&rft.pages=64&rft.edition=illustrated&rft.pub=Cambridge+University+Press&rft.date=1998&rft.isbn=978-0-521-56583-7&rft.aulast=Strange&rft.aufirst=Paul&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsdVrBM2w0Ow&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBallantineDoneganEastham2016" class="citation journal cs1">Ballantine, K. E.; Donegan, J. F.; Eastham, P. R. (2016). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5565928">"There are many ways to spin a photon: Half-quantization of a total optical angular momentum"</a>. Science Advances. 2 (4): e1501748. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016SciA....2E1748B">2016SciA....2E1748B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fsciadv.1501748">10.1126/sciadv.1501748</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5565928">5565928</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28861467">28861467</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+Advances&rft.atitle=There+are+many+ways+to+spin+a+photon%3A+Half-quantization+of+a+total+optical+angular+momentum&rft.volume=2&rft.issue=4&rft.pages=e1501748&rft.date=2016&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5565928%23id-name%3DPMC&rft_id=info%3Apmid%2F28861467&rft_id=info%3Adoi%2F10.1126%2Fsciadv.1501748&rft_id=info%3Abibcode%2F2016SciA....2E1748B&rft.aulast=Ballantine&rft.aufirst=K.+E.&rft.au=Donegan%2C+J.+F.&rft.au=Eastham%2C+P.+R.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5565928&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ._H._Van_Vleck1951" class="citation journal cs1">J. H. Van Vleck (1951). "The Coupling of Angular Momentum Vectors in Molecules". Rev. Mod. Phys. 23 (3): 213. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1951RvMP...23..213V">1951RvMP...23..213V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.23.213">10.1103/RevModPhys.23.213</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rev.+Mod.+Phys.&rft.atitle=The+Coupling+of+Angular+Momentum+Vectors+in+Molecules&rft.volume=23&rft.issue=3&rft.pages=213&rft.date=1951&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.23.213&rft_id=info%3Abibcode%2F1951RvMP...23..213V&rft.au=J.+H.+Van+Vleck&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-littlejohn-43"><a href="#cite_ref-littlejohn_43-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLittlejohn2011" class="citation web cs1"><a href="/wiki/Robert_Grayson_Littlejohn" title="Robert Grayson Littlejohn">Littlejohn, Robert</a> (2011). <a rel="nofollow" class="external text" href="http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf">"Lecture notes on rotations in quantum mechanics"</a> (PDF). Physics 221B Spring 2011. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf">Archived</a> (PDF) from the original on 2022-10-09. Retrieved 13 Jan 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Physics+221B+Spring+2011&rft.atitle=Lecture+notes+on+rotations+in+quantum+mechanics&rft.date=2011&rft.aulast=Littlejohn&rft.aufirst=Robert&rft_id=http%3A%2F%2Fbohr.physics.berkeley.edu%2Fclasses%2F221%2F1011%2Fnotes%2Fspinrot.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-Okulov2008-44"><a href="#cite_ref-Okulov2008_44-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOkulov2008" class="citation journal cs1">Okulov, A Yu (2008). "Angular momentum of photons and phase conjugation". Journal of Physics B: Atomic, Molecular and Optical Physics. 41 (10): 101001. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0801.2675">0801.2675</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008JPhB...41j1001O">2008JPhB...41j1001O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0953-4075%2F41%2F10%2F101001">10.1088/0953-4075/41/10/101001</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13307937">13307937</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics+B%3A+Atomic%2C+Molecular+and+Optical+Physics&rft.atitle=Angular+momentum+of+photons+and+phase+conjugation&rft.volume=41&rft.issue=10&rft.pages=101001&rft.date=2008&rft_id=info%3Aarxiv%2F0801.2675&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13307937%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0953-4075%2F41%2F10%2F101001&rft_id=info%3Abibcode%2F2008JPhB...41j1001O&rft.aulast=Okulov&rft.aufirst=A+Yu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-Okulov2008J-45"><a href="#cite_ref-Okulov2008J_45-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOkulov2008" class="citation journal cs1 cs1-prop-foreign-lang-source">Okulov, A.Y. (2008). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151222101259/http://www.jetpletters.ac.ru/ps/1852/article_28262.shtml">"Optical and Sound Helical structures in a Mandelstam – Brillouin mirror"</a>. JETP Letters (in Russian). 88 (8): 561–566. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008JETPL..88..487O">2008JETPL..88..487O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2Fs0021364008200046">10.1134/s0021364008200046</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120371573">120371573</a>. Archived from <a rel="nofollow" class="external text" href="http://www.jetpletters.ac.ru/ps/1852/article_28262.shtml">the original</a> on 2015-12-22. Retrieved 2015-10-31.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=JETP+Letters&rft.atitle=Optical+and+Sound+Helical+structures+in+a+Mandelstam+%E2%80%93+Brillouin+mirror&rft.volume=88&rft.issue=8&rft.pages=561-566&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120371573%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1134%2Fs0021364008200046&rft_id=info%3Abibcode%2F2008JETPL..88..487O&rft.aulast=Okulov&rft.aufirst=A.Y.&rft_id=http%3A%2F%2Fwww.jetpletters.ac.ru%2Fps%2F1852%2Farticle_28262.shtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewton1803" class="citation book cs1">Newton, Isaac (1803). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=exwAAAAAQAAJ">"Axioms; or Laws of Motion, Law I"</a>. The Mathematical Principles of Natural Philosophy. Andrew Motte, translator. H. D. Symonds, London. p. 322 – via Google books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Axioms%3B+or+Laws+of+Motion%2C+Law+I&rft.btitle=The+Mathematical+Principles+of+Natural+Philosophy&rft.pages=322&rft.pub=H.+D.+Symonds%2C+London&rft.date=1803&rft.aulast=Newton&rft.aufirst=Isaac&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DexwAAAAAQAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> Newton, Axioms; or Laws of Motion, Corollary III </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWithers2013" class="citation journal cs1">Withers, L. P. (2013). <a rel="nofollow" class="external text" href="https://doi.org/10.4169/amer.math.monthly.120.01.071">"Visual Angular Momentum: Mamikon meets Kepler"</a>. American Mathematical Monthly. 120 (1): 71–73. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2Famer.math.monthly.120.01.071">10.4169/amer.math.monthly.120.01.071</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:30994835">30994835</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Visual+Angular+Momentum%3A+Mamikon+meets+Kepler&rft.volume=120&rft.issue=1&rft.pages=71-73&rft.date=2013&rft_id=info%3Adoi%2F10.4169%2Famer.math.monthly.120.01.071&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A30994835%23id-name%3DS2CID&rft.aulast=Withers&rft.aufirst=L.+P.&rft_id=https%3A%2F%2Fdoi.org%2F10.4169%2Famer.math.monthly.120.01.071&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostolMnatsakanian2012" class="citation book cs1">Apostol, Tom M.; Mnatsakanian, Mamikon A. (2012). New Horizons in Geometry. MAA Press. pp. 29–30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-4335-1" title="Special:BookSources/978-1-4704-4335-1"><bdi>978-1-4704-4335-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=New+Horizons+in+Geometry&rft.pages=29-30&rft.pub=MAA+Press&rft.date=2012&rft.isbn=978-1-4704-4335-1&rft.aulast=Apostol&rft.aufirst=Tom+M.&rft.au=Mnatsakanian%2C+Mamikon+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorrelli2011" class="citation web cs1">Borrelli, Arianna (2011). <a rel="nofollow" class="external text" href="http://weatherglass.de/PDFs/Angular_momentum.pdf">"Angular momentum between physics and mathematics"</a> (PDF). <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://weatherglass.de/PDFs/Angular_momentum.pdf">Archived</a> (PDF) from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Angular+momentum+between+physics+and+mathematics&rft.date=2011&rft.aulast=Borrelli&rft.aufirst=Arianna&rft_id=http%3A%2F%2Fweatherglass.de%2FPDFs%2FAngular_momentum.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> for an excellent and detailed summary of the concept of angular momentum through history. </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBruce2008" class="citation web cs1">Bruce, Ian (2008). <a rel="nofollow" class="external text" href="http://www.17centurymaths.com/contents/mechanica1.html">"Euler : Mechanica Vol. 1"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Euler+%3A+Mechanica+Vol.+1&rft.date=2008&rft.aulast=Bruce&rft.aufirst=Ian&rft_id=http%3A%2F%2Fwww.17centurymaths.com%2Fcontents%2Fmechanica1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://eulerarchive.maa.org/correspondence/letters/OO0153.pdf">"Euler's Correspondence with Daniel Bernoulli, Bernoulli to Euler, 04 February, 1744"</a> (PDF). The Euler Archive. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://eulerarchive.maa.org/correspondence/letters/OO0153.pdf">Archived</a> (PDF) from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Euler+Archive&rft.atitle=Euler%27s+Correspondence+with+Daniel+Bernoulli%2C+Bernoulli+to+Euler%2C+04+February%2C+1744&rft_id=http%3A%2F%2Feulerarchive.maa.org%2Fcorrespondence%2Fletters%2FOO0153.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRankine1872" class="citation book cs1">Rankine, W. J. M. (1872). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=u9UKAQAAIAAJ">A Manual of Applied Mechanics</a> (6th ed.). Charles Griffin and Company, London. p. 506 – via Google books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Manual+of+Applied+Mechanics&rft.pages=506&rft.edition=6th&rft.pub=Charles+Griffin+and+Company%2C+London&rft.date=1872&rft.aulast=Rankine&rft.aufirst=W.+J.+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Du9UKAQAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-Hayward-54"><a href="#cite_ref-Hayward_54-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayward1864" class="citation journal cs1">Hayward, Robert B. (1864). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Yx1YAAAAYAAJ">"On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space with Applications"</a>. Transactions of the Cambridge Philosophical Society. 10: 1. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1864TCaPS..10....1H">1864TCaPS..10....1H</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Cambridge+Philosophical+Society&rft.atitle=On+a+Direct+Method+of+estimating+Velocities%2C+Accelerations%2C+and+all+similar+Quantities+with+respect+to+Axes+moveable+in+any+manner+in+Space+with+Applications&rft.volume=10&rft.pages=1&rft.date=1864&rft_id=info%3Abibcode%2F1864TCaPS..10....1H&rft.aulast=Hayward&rft.aufirst=Robert+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYx1YAAAAYAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> see, for instance, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGompertz1818" class="citation journal cs1">Gompertz, Benjamin (1818). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ANE4AAAAMAAJ">"On Pendulums vibrating between Cheeks"</a>. The Journal of Science and the Arts. III (V): 17 – via Google books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Science+and+the+Arts&rft.atitle=On+Pendulums+vibrating+between+Cheeks&rft.volume=III&rft.issue=V&rft.pages=17&rft.date=1818&rft.aulast=Gompertz&rft.aufirst=Benjamin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DANE4AAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988">; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerapath1847" class="citation book cs1">Herapath, John (1847). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nH0tAAAAYAAJ">Mathematical Physics</a>. Whittaker and Co., London. p. 56 – via Google books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Physics&rft.pages=56&rft.pub=Whittaker+and+Co.%2C+London&rft.date=1847&rft.aulast=Herapath&rft.aufirst=John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnH0tAAAAYAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> see, for instance, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLanden1785" class="citation journal cs1">Landen, John (1785). "Of the Rotatory Motion of a Body of any Form whatever". Philosophical Transactions. LXXV (I): 311–332. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1785.0016">10.1098/rstl.1785.0016</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186212814">186212814</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions&rft.atitle=Of+the+Rotatory+Motion+of+a+Body+of+any+Form+whatever&rft.volume=LXXV&rft.issue=I&rft.pages=311-332&rft.date=1785&rft_id=info%3Adoi%2F10.1098%2Frstl.1785.0016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186212814%23id-name%3DS2CID&rft.aulast=Landen&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=36" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen-TannoudjiDiuLaloë2006" class="citation book cs1">Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2006). Quantum Mechanics (2 volume set ed.). John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-56952-7" title="Special:BookSources/978-0-471-56952-7"><bdi>978-0-471-56952-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.edition=2+volume+set&rft.pub=John+Wiley+%26+Sons&rft.date=2006&rft.isbn=978-0-471-56952-7&rft.aulast=Cohen-Tannoudji&rft.aufirst=Claude&rft.au=Diu%2C+Bernard&rft.au=Lalo%C3%AB%2C+Franck&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCondonShortley1935" class="citation book cs1">Condon, E. U.; Shortley, G. H. (1935). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.212979">"Chapter III: Angular Momentum"</a>. The Theory of Atomic Spectra. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-09209-8" title="Special:BookSources/978-0-521-09209-8"><bdi>978-0-521-09209-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+III%3A+Angular+Momentum&rft.btitle=The+Theory+of+Atomic+Spectra&rft.pub=Cambridge+University+Press&rft.date=1935&rft.isbn=978-0-521-09209-8&rft.aulast=Condon&rft.aufirst=E.+U.&rft.au=Shortley%2C+G.+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.212979&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdmonds1957" class="citation book cs1">Edmonds, A. R. (1957). <a rel="nofollow" class="external text" href="https://archive.org/details/angularmomentumi0000edmo">Angular Momentum in Quantum Mechanics</a>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-07912-7" title="Special:BookSources/978-0-691-07912-7"><bdi>978-0-691-07912-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Angular+Momentum+in+Quantum+Mechanics&rft.pub=Princeton+University+Press&rft.date=1957&rft.isbn=978-0-691-07912-7&rft.aulast=Edmonds&rft.aufirst=A.+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fangularmomentumi0000edmo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2013" class="citation cs2">Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013qtm..book.....H">2013qtm..book.....H</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-40122-5" title="Special:BookSources/978-0-387-40122-5"><bdi>978-0-387-40122-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory+for+Mathematicians&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2013&rft_id=info%3Abibcode%2F2013qtm..book.....H&rft.isbn=978-0-387-40122-5&rft.aulast=Hall&rft.aufirst=Brian+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson1998" class="citation book cs1">Jackson, John David (1998). <a rel="nofollow" class="external text" href="https://archive.org/details/classicalelectro0000jack_e8g9">Classical Electrodynamics</a> (3rd ed.). John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-30932-1" title="Special:BookSources/978-0-471-30932-1"><bdi>978-0-471-30932-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Electrodynamics&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=1998&rft.isbn=978-0-471-30932-1&rft.aulast=Jackson&rft.aufirst=John+David&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fclassicalelectro0000jack_e8g9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerwayJewett2004" class="citation book cs1">Serway, Raymond A.; Jewett, John W. (2004). <a rel="nofollow" class="external text" href="https://archive.org/details/physicssciengv2p00serw">Physics for Scientists and Engineers</a> (6th ed.). Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-40842-8" title="Special:BookSources/978-0-534-40842-8"><bdi>978-0-534-40842-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=2004&rft.isbn=978-0-534-40842-8&rft.aulast=Serway&rft.aufirst=Raymond+A.&rft.au=Jewett%2C+John+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fphysicssciengv2p00serw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThompson1994" class="citation book cs1">Thompson, William J. (1994). Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-55264-2" title="Special:BookSources/978-0-471-55264-2"><bdi>978-0-471-55264-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Angular+Momentum%3A+An+Illustrated+Guide+to+Rotational+Symmetries+for+Physical+Systems&rft.pub=Wiley&rft.date=1994&rft.isbn=978-0-471-55264-2&rft.aulast=Thompson&rft.aufirst=William+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTipler2004" class="citation book cs1">Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-0809-4" title="Special:BookSources/978-0-7167-0809-4"><bdi>978-0-7167-0809-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers%3A+Mechanics%2C+Oscillations+and+Waves%2C+Thermodynamics&rft.edition=5th&rft.pub=W.+H.+Freeman&rft.date=2004&rft.isbn=978-0-7167-0809-4&rft.aulast=Tipler&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman_RLeighton_RSands_M2013" class="citation book cs1"><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman R</a>; <a href="/wiki/Robert_B._Leighton" title="Robert B. Leighton">Leighton R</a>; <a href="/wiki/Matthew_Sands" title="Matthew Sands">Sands M</a> (September 2013). <a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_19.html#Ch19-S4">"19–4 Rotational kinetic energy"</a>. <a href="/wiki/The_Feynman_Lectures_on_Physics" title="The Feynman Lectures on Physics">The Feynman Lectures on Physics</a> (online ed.) – via The Feynman Lectures Website.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=19%E2%80%934+Rotational+kinetic+energy&rft.btitle=The+Feynman+Lectures+on+Physics&rft.edition=online&rft.date=2013-09&rft.au=Feynman+R&rft.au=Leighton+R&rft.au=Sands+M&rft_id=https%3A%2F%2Ffeynmanlectures.caltech.edu%2FI_19.html%23Ch19-S4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAngular+momentum" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Angular_momentum&action=edit&section=37" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style 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img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}</style><div role="navigation" aria-labelledby="sister-projects" class="side-box metadata side-box-right sister-box sistersitebox plainlinks"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-abovebelow"> Angular momentum at Wikipedia's <a href="/wiki/Wikipedia:Wikimedia_sister_projects" title="Wikipedia:Wikimedia sister projects">sister projects</a></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/27px-Wiktionary-logo-v2.svg.png" decoding="async" width="27" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/41px-Wiktionary-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/54px-Wiktionary-logo-v2.svg.png 2x" data-file-width="391" data-file-height="391" /><a href="https://en.wiktionary.org/wiki/Special:Search/Angular_momentum" class="extiw" title="wikt:Special:Search/Angular momentum">Definitions</a> from Wiktionary</li><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="20" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /><a href="https://commons.wikimedia.org/wiki/Category:Angular_momentum" class="extiw" title="c:Category:Angular momentum">Media</a> from Commons</li><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/27px-Wikibooks-logo.svg.png" decoding="async" width="27" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/41px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/54px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /><a href="https://en.wikibooks.org/wiki/High_School_Physics" class="extiw" title="b:High School Physics">Textbooks</a> from Wikibooks</li><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png" decoding="async" width="27" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/41px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/54px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /><a href="https://en.wikiversity.org/wiki/Distances/Angular_momenta" class="extiw" title="v:Distances/Angular momenta">Resources</a> from Wikiversity</li><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/27px-Wikidata-logo.svg.png" decoding="async" width="27" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/41px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/54px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /><a href="https://www.wikidata.org/wiki/Q161254" class="extiw" title="d:Q161254">Data</a> from Wikidata</li></ul></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://www.scientificamerican.com/article/football-science-shapes/">"What Do a Submarine, a Rocket and a Football Have in Common?</a> Why the prolate spheroid is the shape for success" (Scientific American, November 8, 2010)</li> <li><a rel="nofollow" class="external text" href="http://www.lightandmatter.com/html_books/lm/ch15/ch15.html">Conservation of Angular Momentum</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161207091606/http://www.lightandmatter.com/html_books/lm/ch15/ch15.html">Archived</a> 2016-12-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> – a chapter from an online textbook</li> <li><a rel="nofollow" class="external text" href="http://www.hakenberg.de/diffgeo/collision_resolution.htm">Angular Momentum in a Collision Process</a> – derivation of the three-dimensional case</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160531064811/https://www.thepracticeset.com/physics-1-angular-momentum-rolling-motion-question-list/">Angular Momentum and Rolling Motion</a> – more momentum theory</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style 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quantities</td> </tr> <tr> <th style="font-weight:normal;font-size:80%;">Dimensions</th> <th style="font-weight:normal;">1</th> <th style="font-weight:normal;">L</th> <th style="font-weight:normal;">L2</th> <th style="font-weight:normal;font-size:80%;">Dimensions</th> <th style="font-weight:normal;">1</th> <th style="font-weight:normal;">θ</th> <th style="font-weight:normal;">θ2</th> </tr> <tr> <th style="font-weight:normal;">T</th> <td><a href="/wiki/Time" title="Time">time</a>: t <a href="/wiki/Second" title="Second">s</a></td> <td><a href="/wiki/Absement" title="Absement">absement</a>: A <a href="/wiki/Meter_second" class="mw-redirect" title="Meter second">m s</a></td> <td></td> <th style="font-weight:normal;">T</th> <td><a href="/wiki/Time" title="Time">time</a>: t <a href="/wiki/Second" title="Second">s</a></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">1</th> <td></td> <td><a href="/wiki/Distance" title="Distance">distance</a>: d, <a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position</a>: r, s, x, <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> <a href="/wiki/Metre" title="Metre">m</a></td> <td><a href="/wiki/Area" title="Area">area</a>: A <a href="/wiki/Square_metre" title="Square metre">m2</a></td> <th style="font-weight:normal;">1</th> <td></td> <td><a href="/wiki/Angle" title="Angle">angle</a>: θ, <a href="/wiki/Angular_displacement" title="Angular displacement">angular displacement</a>: θ <a href="/wiki/Radian" title="Radian">rad</a></td> <td><a href="/wiki/Solid_angle" title="Solid angle">solid angle</a>: Ω <a href="/wiki/Steradian" title="Steradian">rad2, sr</a></td> </tr> <tr> <th style="font-weight:normal;">T−1</th> <td><a href="/wiki/Frequency" title="Frequency">frequency</a>: f <a href="/wiki/Inverse_second" title="Inverse second">s−1</a>, <a href="/wiki/Hertz" title="Hertz">Hz</a></td> <td><a href="/wiki/Speed" title="Speed">speed</a>: v, <a href="/wiki/Velocity" title="Velocity">velocity</a>: v <a href="/wiki/Metre_per_second" title="Metre per second">m s−1</a></td> <td><a href="/wiki/Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a>: ν, <a href="/wiki/Specific_angular_momentum" title="Specific angular momentum">specific angular momentum</a>: h m2 s−1</td> <th style="font-weight:normal;">T−1</th> <td><a href="/wiki/Frequency" title="Frequency">frequency</a>: f, <a href="/wiki/Rotational_speed" class="mw-redirect" title="Rotational speed">rotational speed</a>: n, <a href="/wiki/Rotational_velocity" class="mw-redirect" title="Rotational velocity">rotational velocity</a>: n <a href="/wiki/Inverse_second" title="Inverse second">s−1</a>, <a href="/wiki/Hertz" title="Hertz">Hz</a></td> <td><a href="/wiki/Angular_speed" class="mw-redirect" title="Angular speed">angular speed</a>: ω, <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a>: ω <a href="/wiki/Radian_per_second" title="Radian per second">rad s−1</a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T−2</th> <td></td> <td><a href="/wiki/Acceleration" title="Acceleration">acceleration</a>: a <a href="/wiki/Metre_per_second_squared" title="Metre per second squared">m s−2</a></td> <td></td> <th style="font-weight:normal;">T−2</th> <td><a href="/wiki/Rotational_acceleration" class="mw-redirect" title="Rotational acceleration">rotational acceleration</a> <a href="/wiki/Inverse_square_second" class="mw-redirect" title="Inverse square second">s−2</a></td> <td><a href="/wiki/Angular_acceleration" title="Angular acceleration">angular acceleration</a>: α <a href="/wiki/Radian_per_second_squared" class="mw-redirect" title="Radian per second squared">rad s−2</a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T−3</th> <td></td> <td><a href="/wiki/Jerk_(physics)" title="Jerk (physics)">jerk</a>: j m s−3</td> <td></td> <th style="font-weight:normal;">T−3</th> <td></td> <td><a href="/wiki/Jerk_(physics)#Jerk_in_rotation" title="Jerk (physics)">angular jerk</a>: ζ rad s−3</td> <td></td> </tr> <tr style="border-top: 3px double #a2a9b1;"> <th style="font-weight:normal;">M</th> <td><a href="/wiki/Mass" title="Mass">mass</a>: m <a href="/wiki/Kilogram" title="Kilogram">kg</a></td> <td>weighted position: M ⟨x⟩ = ∑ m x </td> <td></td> <th style="font-weight:normal;">ML2</th> <td><a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>: I <a href="/wiki/Kilogram_square_metre" class="mw-redirect" title="Kilogram square metre">kg m2</a></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">MT−1</th> <td><a href="/wiki/Mass_flow_rate" title="Mass flow rate">Mass flow rate</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad59b9876301e8fb75b9ddbf08de594b87251d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:2.176ex;" alt="{\displaystyle {\dot {m}}}"> <a href="/wiki/Kilogram_per_second" class="mw-redirect" title="Kilogram per second">kg s−1</a></td> <td><a href="/wiki/Momentum" title="Momentum">momentum</a>: p, <a href="/wiki/Impulse_(physics)" title="Impulse (physics)">impulse</a>: J <a href="/wiki/Kilogram_metre_per_second" class="mw-redirect" title="Kilogram metre per second">kg m s−1</a>, <a href="/wiki/Newton_second" class="mw-redirect" title="Newton second">N s</a></td> <td><a href="/wiki/Action_(physics)" title="Action (physics)">action</a>: 𝒮, <a href="/wiki/Absement#Applications" title="Absement">actergy</a>: ℵ <a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg m2 s−1</a>, <a href="/wiki/Joule-second" title="Joule-second">J s</a></td> <th style="font-weight:normal;">ML2T−1</th> <td></td> <td><a class="mw-selflink selflink">angular momentum</a>: L, <a href="/wiki/List_of_equations_in_classical_mechanics#Derived_dynamic_quantities" title="List of equations in classical mechanics">angular impulse</a>: ΔL <a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg m2 s−1</a></td> <td><a href="/wiki/Action_(physics)" title="Action (physics)">action</a>: 𝒮, <a href="/wiki/Absement#Applications" title="Absement">actergy</a>: ℵ <a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg m2 s−1</a>, <a href="/wiki/Joule-second" title="Joule-second">J s</a></td> </tr> <tr> <th style="font-weight:normal;">MT−2</th> <td></td> <td><a href="/wiki/Force" title="Force">force</a>: F, <a href="/wiki/Weight" title="Weight">weight</a>: Fg kg m s−2, <a href="/wiki/Newton_(unit)" title="Newton (unit)">N</a></td> <td><a href="/wiki/Energy" title="Energy">energy</a>: E, <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>: W, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a>: L kg m2 s−2, <a href="/wiki/Joule" title="Joule">J</a></td> <th style="font-weight:normal;">ML2T−2</th> <td></td> <td><a href="/wiki/Torque" title="Torque">torque</a>: τ, <a href="/wiki/Torque#Terminology" title="Torque">moment</a>: M kg m2 s−2, <a href="/wiki/Newton-metre" title="Newton-metre">N m</a></td> <td><a href="/wiki/Energy" title="Energy">energy</a>: E, <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>: W, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a>: L kg m2 s−2, <a href="/wiki/Joule" title="Joule">J</a></td> </tr> <tr> <th style="font-weight:normal;">MT−3</th> <td></td> <td><a href="/wiki/Yank_(physics)" class="mw-redirect" title="Yank (physics)">yank</a>: Y kg m s−3, N s−1</td> <td><a href="/wiki/Power_(physics)" title="Power (physics)">power</a>: P kg m2 s−3, <a href="/wiki/Watt" title="Watt">W</a></td> <th style="font-weight:normal;">ML2T−3</th> <td></td> <td><a href="/wiki/Rotatum" class="mw-redirect" title="Rotatum">rotatum</a>: P kg m2 s−3, N m s−1</td> <td><a href="/wiki/Power_(physics)" title="Power (physics)">power</a>: P kg m2 s−3, <a href="/wiki/Watt" title="Watt">W</a></td> </tr> </tbody></table></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Tensors" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Tensors" title="Template:Tensors"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tensors" title="Template talk:Tensors"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tensors" title="Special:EditPage/Template:Tensors"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Tensors" style="font-size:114%;margin:0 4em"><a href="/wiki/Tensor" title="Tensor">Tensors</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="/wiki/Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scope</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coordinate_system" title="Coordinate system">Coordinate system</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li> <li><a href="/wiki/Dyadics" title="Dyadics">Dyadic algebra</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="/wiki/Physics" title="Physics">Physics</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li> <li><a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a></li> <li><a href="/wiki/Index_notation" title="Index notation">Index notation</a></li> <li><a href="/wiki/Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li> <li><a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li> <li><a href="/wiki/Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li> <li><a href="/wiki/Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li> <li><a href="/wiki/Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li> <li><a href="/wiki/Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor definitions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a></li> <li><a href="/wiki/Tensor_density" title="Tensor density">Tensor density</a></li> <li><a href="/wiki/Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li> <li><a href="/wiki/Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li> <li><a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li> <li><a href="/wiki/Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li> <li><a href="/wiki/Tensor_operator" title="Tensor operator">Tensor operator</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li> <li><a href="/wiki/Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li> <li><a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a></li> <li><a href="/wiki/Symmetrization" title="Symmetrization">Symmetrization</a></li> <li><a href="/wiki/Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine connection</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Connection form</a></li> <li><a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Dimension" title="Dimension">Dimension</a></li> <li><a href="/wiki/Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Pseudotensor" title="Pseudotensor">Pseudotensor</a></li> <li><a href="/wiki/Spinor" title="Spinor">Spinor</a></li> <li><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li> <li><a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a></li> <li><a href="/wiki/Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li> <li><a href="/wiki/Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li> <li><a class="mw-selflink-fragment" href="#Angular_momentum_in_relativistic_mechanics">Angular momentum tensor</a></li> <li><a href="/wiki/Spin_tensor" title="Spin tensor">Spin tensor</a></li> <li><a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li> <li><a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li> <li><a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li> <li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li> <li><a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li> <li><a href="/wiki/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li> <li><a href="/wiki/Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li> <li><a href="/wiki/Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" 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