Lagrangian mechanics

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</li> <li id="toc-From_Newtonian_to_Lagrangian_mechanics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#From_Newtonian_to_Lagrangian_mechanics"> <div class="vector-toc-text"> 2 From Newtonian to Lagrangian mechanics </div> </a> <button aria-controls="toc-From_Newtonian_to_Lagrangian_mechanics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle From Newtonian to Lagrangian mechanics subsection </button> <ul id="toc-From_Newtonian_to_Lagrangian_mechanics-sublist" class="vector-toc-list"> <li id="toc-Newton's_laws" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Newton's_laws"> <div class="vector-toc-text"> 2.1 Newton's laws </div> </a> <ul id="toc-Newton's_laws-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-D'Alembert's_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#D'Alembert's_principle"> <div class="vector-toc-text"> 2.2 D'Alembert's principle </div> </a> <ul id="toc-D'Alembert's_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equations_of_motion_from_D'Alembert's_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equations_of_motion_from_D'Alembert's_principle"> <div class="vector-toc-text"> 2.3 Equations of motion from D'Alembert's principle </div> </a> <ul id="toc-Equations_of_motion_from_D'Alembert's_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euler–Lagrange_equations_and_Hamilton's_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euler–Lagrange_equations_and_Hamilton's_principle"> <div class="vector-toc-text"> 2.4 Euler–Lagrange equations and Hamilton's principle </div> </a> <ul id="toc-Euler–Lagrange_equations_and_Hamilton's_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lagrange_multipliers_and_constraints" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lagrange_multipliers_and_constraints"> <div class="vector-toc-text"> 2.5 Lagrange multipliers and constraints </div> </a> <ul id="toc-Lagrange_multipliers_and_constraints-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties_of_the_Lagrangian" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties_of_the_Lagrangian"> <div class="vector-toc-text"> 3 Properties of the Lagrangian </div> </a> <button aria-controls="toc-Properties_of_the_Lagrangian-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties of the Lagrangian subsection </button> <ul id="toc-Properties_of_the_Lagrangian-sublist" class="vector-toc-list"> <li id="toc-Non-uniqueness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-uniqueness"> <div class="vector-toc-text"> 3.1 Non-uniqueness </div> </a> <ul id="toc-Non-uniqueness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Invariance_under_point_transformations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Invariance_under_point_transformations"> <div class="vector-toc-text"> 3.2 Invariance under point transformations </div> </a> <ul id="toc-Invariance_under_point_transformations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cyclic_coordinates_and_conserved_momenta" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cyclic_coordinates_and_conserved_momenta"> <div class="vector-toc-text"> 3.3 Cyclic coordinates and conserved momenta </div> </a> <ul id="toc-Cyclic_coordinates_and_conserved_momenta-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Energy"> <div class="vector-toc-text"> 3.4 Energy </div> </a> <ul id="toc-Energy-sublist" class="vector-toc-list"> <li id="toc-Invariance_under_coordinate_transformations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Invariance_under_coordinate_transformations"> <div class="vector-toc-text"> 3.4.1 Invariance under coordinate transformations </div> </a> <ul id="toc-Invariance_under_coordinate_transformations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conservation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conservation"> <div class="vector-toc-text"> 3.4.2 Conservation </div> </a> <ul id="toc-Conservation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kinetic_and_potential_energies" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Kinetic_and_potential_energies"> <div class="vector-toc-text"> 3.4.3 Kinetic and potential energies </div> </a> <ul id="toc-Kinetic_and_potential_energies-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mechanical_similarity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mechanical_similarity"> <div class="vector-toc-text"> 3.5 Mechanical similarity </div> </a> <ul id="toc-Mechanical_similarity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interacting_particles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interacting_particles"> <div class="vector-toc-text"> 3.6 Interacting particles </div> </a> <ul id="toc-Interacting_particles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Consequences_of_singular_Lagrangians" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Consequences_of_singular_Lagrangians"> <div class="vector-toc-text"> 3.7 Consequences of singular Lagrangians </div> </a> <ul id="toc-Consequences_of_singular_Lagrangians-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 4 Examples </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Examples subsection </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Conservative_force" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conservative_force"> <div class="vector-toc-text"> 4.1 Conservative force </div> </a> <ul id="toc-Conservative_force-sublist" class="vector-toc-list"> <li id="toc-Cartesian_coordinates" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cartesian_coordinates"> <div class="vector-toc-text"> 4.1.1 Cartesian coordinates </div> </a> <ul id="toc-Cartesian_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polar_coordinates_in_2D_and_3D" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Polar_coordinates_in_2D_and_3D"> <div class="vector-toc-text"> 4.1.2 Polar coordinates in 2D and 3D </div> </a> <ul id="toc-Polar_coordinates_in_2D_and_3D-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pendulum_on_a_movable_support" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pendulum_on_a_movable_support"> <div class="vector-toc-text"> 4.2 Pendulum on a movable support </div> </a> <ul id="toc-Pendulum_on_a_movable_support-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Two-body_central_force_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two-body_central_force_problem"> <div class="vector-toc-text"> 4.3 Two-body central force problem </div> </a> <ul id="toc-Two-body_central_force_problem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions_to_include_non-conservative_forces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Extensions_to_include_non-conservative_forces"> <div class="vector-toc-text"> 5 Extensions to include non-conservative forces </div> </a> <button aria-controls="toc-Extensions_to_include_non-conservative_forces-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Extensions to include non-conservative forces subsection </button> <ul id="toc-Extensions_to_include_non-conservative_forces-sublist" class="vector-toc-list"> <li id="toc-Dissipative_forces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dissipative_forces"> <div class="vector-toc-text"> 5.1 Dissipative forces </div> </a> <ul id="toc-Dissipative_forces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electromagnetism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electromagnetism"> <div class="vector-toc-text"> 5.2 Electromagnetism </div> </a> <ul id="toc-Electromagnetism-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_contexts_and_formulations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_contexts_and_formulations"> <div class="vector-toc-text"> 6 Other contexts and formulations </div> </a> <button aria-controls="toc-Other_contexts_and_formulations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Other contexts and formulations subsection </button> <ul id="toc-Other_contexts_and_formulations-sublist" class="vector-toc-list"> <li id="toc-Alternative_formulations_of_classical_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternative_formulations_of_classical_mechanics"> <div class="vector-toc-text"> 6.1 Alternative formulations of classical mechanics </div> </a> <ul id="toc-Alternative_formulations_of_classical_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Momentum_space_formulation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Momentum_space_formulation"> <div class="vector-toc-text"> 6.2 Momentum space formulation </div> </a> <ul id="toc-Momentum_space_formulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher_derivatives_of_generalized_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher_derivatives_of_generalized_coordinates"> <div class="vector-toc-text"> 6.3 Higher derivatives of generalized coordinates </div> </a> <ul id="toc-Higher_derivatives_of_generalized_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optics"> <div class="vector-toc-text"> 6.4 Optics </div> </a> <ul id="toc-Optics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_formulation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic_formulation"> <div class="vector-toc-text"> 6.5 Relativistic formulation </div> </a> <ul id="toc-Relativistic_formulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_mechanics"> <div class="vector-toc-text"> 6.6 Quantum mechanics </div> </a> <ul id="toc-Quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classical_field_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_field_theory"> <div class="vector-toc-text"> 6.7 Classical field theory </div> </a> <ul id="toc-Classical_field_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Noether's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noether's_theorem"> <div class="vector-toc-text"> 6.8 Noether's theorem </div> </a> <ul id="toc-Noether's_theorem-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"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> 8 Footnotes </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 9 Notes </div> </a> <ul id="toc-Notes-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"> 10 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"> 11 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"> 12 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">Lagrangian mechanics</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 40 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-40" aria-hidden="true" > 40 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/Lagrange-meganika" title="Lagrange-meganika – Afrikaans" lang="af" hreflang="af" data-title="Lagrange-meganika" 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/%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D9%84%D8%A7%D8%BA%D8%B1%D8%A7%D9%86%D8%AC" title="ميكانيكا لاغرانج – Arabic" lang="ar" hreflang="ar" data-title="ميكانيكا لاغرانج" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%B0%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Лагранжава механіка – Belarusian" lang="be" hreflang="be" data-title="Лагранжава механіка" 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%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0_%D0%BD%D0%B0_%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6" title="Механика на Лагранж – Bulgarian" lang="bg" hreflang="bg" data-title="Механика на Лагранж" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Formulaci%C3%B3_lagrangiana" title="Formulació lagrangiana – Catalan" lang="ca" hreflang="ca" data-title="Formulació lagrangiana" 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%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%BB%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Лагранжла механика – Chuvash" lang="cv" hreflang="cv" data-title="Лагранжла механика" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lagrange-Formalismus" title="Lagrange-Formalismus – German" lang="de" hreflang="de" data-title="Lagrange-Formalismus" 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/Lagrange%27i_mehaanika" title="Lagrange'i mehaanika – Estonian" lang="et" hreflang="et" data-title="Lagrange'i mehaanika" 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%9B%CE%B1%CE%B3%CE%BA%CF%81%CE%B1%CE%BD%CE%B6%CE%B9%CE%B1%CE%BD%CE%AE_%CE%BC%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE" title="Λαγκρανζιανή μηχανική – Greek" lang="el" hreflang="el" data-title="Λαγκρανζιανή μηχανική" 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/Mec%C3%A1nica_lagrangiana" title="Mecánica lagrangiana – Spanish" lang="es" hreflang="es" data-title="Mecánica lagrangiana" 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/Lagran%C4%9Da_mekaniko" title="Lagranĝa mekaniko – Esperanto" lang="eo" hreflang="eo" data-title="Lagranĝa mekaniko" 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/Lagrangeren_mekanika" title="Lagrangeren mekanika – Basque" lang="eu" hreflang="eu" data-title="Lagrangeren mekanika" 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/%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%D9%84%D8%A7%DA%AF%D8%B1%D8%A7%D9%86%DA%98%DB%8C" title="مکانیک لاگرانژی – Persian" lang="fa" hreflang="fa" data-title="مکانیک لاگرانژی" 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/%C3%89quations_de_Lagrange" title="Équations de Lagrange – French" lang="fr" hreflang="fr" data-title="Équations de Lagrange" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Mec%C3%A1nica_lagranxiana" title="Mecánica lagranxiana – Galician" lang="gl" hreflang="gl" data-title="Mecánica lagranxiana" 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/%EB%9D%BC%EA%B7%B8%EB%9E%91%EC%A3%BC_%EC%97%AD%ED%95%99" title="라그랑주 역학 – Korean" lang="ko" hreflang="ko" data-title="라그랑주 역학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B2%E0%A4%BE%E0%A4%97%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%82%E0%A4%9C%E0%A5%80%E0%A4%AF_%E0%A4%AF%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80" title="लाग्रांजीय यांत्रिकी – Hindi" lang="hi" hreflang="hi" data-title="लाग्रांजीय यांत्रिकी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Mekanika_Lagrangian" title="Mekanika Lagrangian – Indonesian" lang="id" hreflang="id" data-title="Mekanika Lagrangian" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Meccanica_lagrangiana" title="Meccanica lagrangiana – Italian" lang="it" hreflang="it" data-title="Meccanica lagrangiana" 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%9E%D7%9B%D7%A0%D7%99%D7%A7%D7%94_%D7%9C%D7%92%D7%A8%D7%90%D7%A0%D7%96%27%D7%99%D7%AA" title="מכניקה לגראנז'ית – Hebrew" lang="he" hreflang="he" data-title="מכניקה לגראנז'ית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv 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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 mw-collapsed"><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 href="/wiki/Angular_momentum" title="Angular momentum">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"><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 class="mw-selflink selflink">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" 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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> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Lagrange_portrait.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Lagrange_portrait.jpg/150px-Lagrange_portrait.jpg" decoding="async" width="150" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Lagrange_portrait.jpg/225px-Lagrange_portrait.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/1/19/Lagrange_portrait.jpg 2x" data-file-width="235" data-file-height="264" /></a><figcaption><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> (1736–1813)</figcaption></figure> In <a href="/wiki/Physics" title="Physics">physics</a>, Lagrangian mechanics is a formulation of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> founded on the <a href="/wiki/Stationary-action_principle" class="mw-redirect" title="Stationary-action principle">stationary-action principle</a> (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> in his presentation to the Turin Academy of Science in 1760<a href="#cite_note-1">[1]</a> culminating in his 1788 grand opus, <a href="/wiki/M%C3%A9canique_analytique" title="Mécanique analytique">Mécanique analytique</a>.<a href="#cite_note-2">[2]</a> Lagrangian mechanics describes a mechanical system as a pair (M, L) consisting of a <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a> M and a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb88de7e4d31737dae8f02575033272f29e6720" 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="{\textstyle L}"> within that space called a Lagrangian. For many systems, L = T − V, where T and V are the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic</a> and <a href="/wiki/Potential_energy" title="Potential energy">potential</a> energy of the system, respectively.<a href="#cite_note-3">[3]</a> The stationary action principle requires that the <a href="/wiki/Action_(physics)#Action_(functional)" title="Action (physics)">action functional</a> of the system derived from L must remain at a stationary point (a <a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">maximum</a>, <a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">minimum</a>, or <a href="/wiki/Saddle_point" title="Saddle point">saddle</a>) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.<a href="#cite_note-4">[4]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=1" title="Edit section: Introduction">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bead_on_wire_constraint.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Bead_on_wire_constraint.svg/220px-Bead_on_wire_constraint.svg.png" decoding="async" width="220" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Bead_on_wire_constraint.svg/330px-Bead_on_wire_constraint.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Bead_on_wire_constraint.svg/440px-Bead_on_wire_constraint.svg.png 2x" data-file-width="345" data-file-height="245" /></a><figcaption>Bead constrained to move on a frictionless wire. The wire exerts a reaction force C on the bead to keep it on the wire. The non-constraint force N in this case is gravity. Notice the initial position of the bead on the wire can lead to different motions.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pendulum_constraint.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Pendulum_constraint.svg/150px-Pendulum_constraint.svg.png" decoding="async" width="150" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Pendulum_constraint.svg/225px-Pendulum_constraint.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Pendulum_constraint.svg/300px-Pendulum_constraint.svg.png 2x" data-file-width="207" data-file-height="269" /></a><figcaption>Simple pendulum. Since the rod is rigid, the position of the bob is constrained according to the equation f(x, y) = 0, the constraint force C is the tension in the rod. Again the non-constraint force N in this case is gravity.</figcaption></figure> Newton's laws and the concept of forces are the usual starting point for teaching about mechanical systems.<a href="#cite_note-Ball2019-5">[5]</a> This method works well for many problems, but for others the approach is nightmarishly complicated.<a href="#cite_note-TatumClassNotes-6">[6]</a> For example, in calculation of the motion of a <a href="/wiki/Torus" title="Torus">torus</a> rolling on a horizontal surface with a pearl sliding inside, the time-varying constraint forces like the <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> of the torus, motion of the pearl in relation to the torus made it difficult to determine the motion of the torus with Newton's equations.<a href="#cite_note-:0-7">[7]</a> Lagrangian mechanics adopts energy rather than force as its basic ingredient,<a href="#cite_note-Ball2019-5">[5]</a> leading to more abstract equations capable of tackling more complex problems.<a href="#cite_note-TatumClassNotes-6">[6]</a> Particularly, Lagrange's approach was to set up independent <a href="/wiki/Generalized_coordinate" class="mw-redirect" title="Generalized coordinate">generalized coordinates</a> for the position and speed of every object, which allows the writing down of a general form of lagrangian (total kinetic energy minus potential energy of the system) and summing this over all possible paths of motion of the particles yielded a formula for the 'action', which he minimized to give a generalized set of equations. This summed quantity is minimized along the path that the particle actually takes. This choice eliminates the need for the constraint force to enter into the resultant generalized <a href="/wiki/System_of_equations" title="System of equations">system of equations</a>. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment.<a href="#cite_note-:0-7">[7]</a> For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a <a href="/wiki/Point_particle" title="Point particle">point particle</a>. For a system of N point particles with <a href="/wiki/Mass" title="Mass">masses</a> m1, m2, ..., mN, each particle has a <a href="/wiki/Position_vector" class="mw-redirect" title="Position vector">position vector</a>, denoted r1, r2, ..., rN. <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> are often sufficient, so r1 = (x1, y1, z1), r2 = (x2, y2, z2) and so on. In <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>, each position vector requires three <a href="/wiki/Coordinate_vector" title="Coordinate vector">coordinates</a> to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system. These are all specific points in space to locate the particles; a general point in space is written r = (x, y, z). The <a href="/wiki/Velocity" title="Velocity">velocity</a> of each particle is how fast the particle moves along its path of motion, and is the <a href="/wiki/Time_derivative" title="Time derivative">time derivative</a> of its position, thus <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1}={\frac {d\mathbf {r} _{1}}{dt}},\mathbf {v} _{2}={\frac {d\mathbf {r} _{2}}{dt}},\ldots ,\mathbf {v} _{N}={\frac {d\mathbf {r} _{N}}{dt}}.}"> <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"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{1}={\frac {d\mathbf {r} _{1}}{dt}},\mathbf {v} _{2}={\frac {d\mathbf {r} _{2}}{dt}},\ldots ,\mathbf {v} _{N}={\frac {d\mathbf {r} _{N}}{dt}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ef980cc96dbae584fbad337cf3ecbf80c71ffc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.449ex; height:5.509ex;" alt="{\displaystyle \mathbf {v} _{1}={\frac {d\mathbf {r} _{1}}{dt}},\mathbf {v} _{2}={\frac {d\mathbf {r} _{2}}{dt}},\ldots ,\mathbf {v} _{N}={\frac {d\mathbf {r} _{N}}{dt}}.}"> In Newtonian mechanics, the <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a> are given by <a href="/wiki/Newton%27s_laws" class="mw-redirect" title="Newton's laws">Newton's laws</a>. The second law "net <a href="/wiki/Force" title="Force">force</a> equals mass times <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>", <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum \mathbf {F} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum \mathbf {F} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245238ffac44917a07f572330afb946161e1dc28" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.421ex; height:6.009ex;" alt="{\displaystyle \sum \mathbf {F} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}},}"> applies to each particle. For an N-particle system in 3 dimensions, there are 3N second-order <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> in the positions of the particles to solve for. <div class="mw-heading mw-heading3"><h3 id="Lagrangian">Lagrangian</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=2" title="Edit section: Lagrangian">edit</a>]</div> Instead of forces, Lagrangian mechanics uses the <a href="/wiki/Energy" title="Energy">energies</a> in the system. The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The non-relativistic Lagrangian for a system of particles in the absence of an electromagnetic field is given by<a href="#cite_note-8">[8]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=T-V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>T</mi> <mo>−</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=T-V,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565eceeb4f59399906dfdd530fade115464701c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.592ex; height:2.509ex;" alt="{\displaystyle L=T-V,}"> where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}v_{k}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}v_{k}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5da405e924d28d8abf11a5d7da72db2c24b5fef2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.208ex; height:7.343ex;" alt="{\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}v_{k}^{2}}"> is the total <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> of the system, equaling the <a href="/wiki/Summation" title="Summation">sum</a> Σ of the kinetic energies of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> particles. Each particle labeled <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}"> has mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b61e803b12f284f3dbda131c190d358c305e72a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.776ex; height:2.009ex;" alt="{\displaystyle m_{k},}"> and vk2 = vk · vk is the magnitude squared of its velocity, equivalent to the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of the velocity with itself.<a href="#cite_note-Torby_1984_page=269-9">[9]</a> Kinetic energy T is the energy of the system's motion and is a function only of the velocities vk, not the positions rk, nor time t, so T = T(v1, v2, ...). V, the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> of the system, reflects the energy of interaction between the particles, i.e. how much energy any one particle has due to all the others, together with any external influences. For <a href="/wiki/Conservative_force" title="Conservative force">conservative forces</a> (e.g. <a href="/wiki/Newtonian_gravity" class="mw-redirect" title="Newtonian gravity">Newtonian gravity</a>), it is a function of the position vectors of the particles only, so V = V(r1, r2, ...). For those non-conservative forces which can be derived from an appropriate potential (e.g. <a href="/wiki/Electromagnetic_potential" class="mw-redirect" title="Electromagnetic potential">electromagnetic potential</a>), the velocities will appear also, V = V(r1, r2, ..., v1, v2, ...). If there is some external field or external driving force changing with time, the potential changes with time, so most generally V = V(r1, r2, ..., v1, v2, ..., t). As already noted, this form of L is applicable to many important classes of system, but not everywhere. For <a href="/wiki/Relativistic_Lagrangian_mechanics" title="Relativistic Lagrangian mechanics">relativistic Lagrangian mechanics</a> it must be replaced as a whole by a function consistent with special relativity (scalar under Lorentz transformations) or general relativity (4-scalar).<a href="#cite_note-10">[10]</a> Where a magnetic field is present, the expression for the potential energy needs restating.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] And for dissipative forces (e.g., <a href="/wiki/Friction" title="Friction">friction</a>), another function must be introduced alongside Lagrangian often referred to as a "Rayleigh dissipation function" to account for the loss of energy.<a href="#cite_note-11">[11]</a> One or more of the particles may each be subject to one or more <a href="/wiki/Holonomic_constraints" title="Holonomic constraints">holonomic constraints</a>; such a constraint is described by an equation of the form f(r, t) = 0. If the number of constraints in the system is C, then each constraint has an equation f1(r, t) = 0, f2(r, t) = 0, ..., fC(r, t) = 0, each of which could apply to any of the particles. If particle k is subject to constraint i, then fi(rk, t) = 0. At any instant of time, the coordinates of a constrained particle are linked together and not independent. The constraint equations determine the allowed paths the particles can move along, but not where they are or how fast they go at every instant of time. <a href="/wiki/Nonholonomic_constraints" class="mw-redirect" title="Nonholonomic constraints">Nonholonomic constraints</a> depend on the particle velocities, accelerations, or higher derivatives of position. Lagrangian mechanics can only be applied to systems whose constraints, if any, are all holonomic. Three examples of nonholonomic constraints are:<a href="#cite_note-12">[12]</a> when the constraint equations are non-integrable, when the constraints have inequalities, or when the constraints involve complicated non-conservative forces like friction. Nonholonomic constraints require special treatment, and one may have to revert to <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newtonian mechanics</a> or use other methods.<a href="#cite_note-13">[13]</a> If T or V or both depend explicitly on time due to time-varying constraints or external influences, the Lagrangian L(r1, r2, ... v1, v2, ... t) is explicitly time-dependent. If neither the potential nor the kinetic energy depend on time, then the Lagrangian L(r1, r2, ... v1, v2, ...) is explicitly independent of time. In either case, the Lagrangian always has implicit time dependence through the generalized coordinates. With these definitions, Lagrange's equations of the first kind are<a href="#cite_note-14">[14]</a> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">Lagrange's equations (first kind) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1be2c3836380f27d95f9006a0d08d7dcd367cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.549ex; height:7.343ex;" alt="{\displaystyle {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}"> </div> where k = 1, 2, ..., N labels the particles, there is a <a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a> λi for each constraint equation fi, and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial \mathbf {r} _{k}}}\equiv \left({\frac {\partial }{\partial x_{k}}},{\frac {\partial }{\partial y_{k}}},{\frac {\partial }{\partial z_{k}}}\right),\quad {\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}\equiv \left({\frac {\partial }{\partial {\dot {x}}_{k}}},{\frac {\partial }{\partial {\dot {y}}_{k}}},{\frac {\partial }{\partial {\dot {z}}_{k}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>≡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>≡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial \mathbf {r} _{k}}}\equiv \left({\frac {\partial }{\partial x_{k}}},{\frac {\partial }{\partial y_{k}}},{\frac {\partial }{\partial z_{k}}}\right),\quad {\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}\equiv \left({\frac {\partial }{\partial {\dot {x}}_{k}}},{\frac {\partial }{\partial {\dot {y}}_{k}}},{\frac {\partial }{\partial {\dot {z}}_{k}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68549b30f49d42b82c9dc423745020eabdffa36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.605ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial }{\partial \mathbf {r} _{k}}}\equiv \left({\frac {\partial }{\partial x_{k}}},{\frac {\partial }{\partial y_{k}}},{\frac {\partial }{\partial z_{k}}}\right),\quad {\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}\equiv \left({\frac {\partial }{\partial {\dot {x}}_{k}}},{\frac {\partial }{\partial {\dot {y}}_{k}}},{\frac {\partial }{\partial {\dot {z}}_{k}}}\right)}"> are each shorthands for a vector of <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> ∂/∂ with respect to the indicated variables (not a derivative with respect to the entire vector).<a href="#cite_note-15">[nb 1]</a> Each overdot is a shorthand for a <a href="/wiki/Time_derivative" title="Time derivative">time derivative</a>. This procedure does increase the number of equations to solve compared to Newton's laws, from 3N to 3N + C, because there are 3N coupled second-order differential equations in the position coordinates and multipliers, plus C constraint equations. However, when solved alongside the position coordinates of the particles, the multipliers can yield information about the constraint forces. The coordinates do not need to be eliminated by solving the constraint equations. In the Lagrangian, the position coordinates and velocity components are all <a href="/wiki/Dependent_and_independent_variables#In_pure_mathematics" title="Dependent and independent variables">independent variables</a>, and derivatives of the Lagrangian are taken with respect to these separately according to the usual <a href="/wiki/Differentiation_rules" title="Differentiation rules">differentiation rules</a> (e.g. the partial derivative of L with respect to the z velocity component of particle 2, defined by vz,2 = dz2/dt, is just ∂L/∂vz,2; no awkward <a href="/wiki/Chain_rule" title="Chain rule">chain rules</a> or total derivatives need to be used to relate the velocity component to the corresponding coordinate z2). In each constraint equation, one coordinate is redundant because it is determined from the other coordinates. The number of independent coordinates is therefore n = 3N − C. We can transform each position vector to a common set of n <a href="/wiki/Generalized_coordinates" title="Generalized coordinates">generalized coordinates</a>, conveniently written as an n-tuple q = (q1, q2, ... qn), by expressing each position vector, and hence the position coordinates, as <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> of the generalized coordinates and time: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{k}=\mathbf {r} _{k}(\mathbf {q} ,t)={\big (}x_{k}(\mathbf {q} ,t),y_{k}(\mathbf {q} ,t),z_{k}(\mathbf {q} ,t),t{\big )}.}"> <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>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{k}=\mathbf {r} _{k}(\mathbf {q} ,t)={\big (}x_{k}(\mathbf {q} ,t),y_{k}(\mathbf {q} ,t),z_{k}(\mathbf {q} ,t),t{\big )}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38e7aa8f2bcad6ededfd09177059750ff1282649" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.507ex; height:3.176ex;" alt="{\displaystyle \mathbf {r} _{k}=\mathbf {r} _{k}(\mathbf {q} ,t)={\big (}x_{k}(\mathbf {q} ,t),y_{k}(\mathbf {q} ,t),z_{k}(\mathbf {q} ,t),t{\big )}.}"> The vector q is a point in the <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a> of the system. The time derivatives of the generalized coordinates are called the generalized velocities, and for each particle the transformation of its velocity vector, the <a href="/wiki/Total_derivative" title="Total derivative">total derivative</a> of its position with respect to time, is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {q}}_{j}={\frac {\mathrm {d} q_{j}}{\mathrm {d} t}},\quad \mathbf {v} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}{\dot {q}}_{j}+{\frac {\partial \mathbf {r} _{k}}{\partial t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {q}}_{j}={\frac {\mathrm {d} q_{j}}{\mathrm {d} t}},\quad \mathbf {v} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}{\dot {q}}_{j}+{\frac {\partial \mathbf {r} _{k}}{\partial t}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd8293f4e12305ea302bd07da731f0ca1fa45a59" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.622ex; height:7.176ex;" alt="{\displaystyle {\dot {q}}_{j}={\frac {\mathrm {d} q_{j}}{\mathrm {d} t}},\quad \mathbf {v} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}{\dot {q}}_{j}+{\frac {\partial \mathbf {r} _{k}}{\partial t}}.}"> Given this vk, the kinetic energy in generalized coordinates depends on the generalized velocities, generalized coordinates, and time if the position vectors depend explicitly on time due to time-varying constraints, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=T(\mathbf {q} ,{\dot {\mathbf {q} }},t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=T(\mathbf {q} ,{\dot {\mathbf {q} }},t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477e2c42e1a50c869d3a6c3a91baf04892255289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.566ex; height:2.843ex;" alt="{\displaystyle T=T(\mathbf {q} ,{\dot {\mathbf {q} }},t).}"> With these definitions, the <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equations</a>, or Lagrange's equations of the second kind<a href="#cite_note-16">[15]</a><a href="#cite_note-17">[16]</a><a href="#cite_note-18">[17]</a> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">Lagrange's equations (second kind) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)={\frac {\partial L}{\partial q_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)={\frac {\partial L}{\partial q_{j}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9f84b11bc8963b63f3775c2e268d8d9cde77e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.677ex; height:7.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)={\frac {\partial L}{\partial q_{j}}}}"> </div> are mathematical results from the <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a>, which can also be used in mechanics. Substituting in the Lagrangian L(q, dq/dt, t) gives the <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a> of the system. The number of equations has decreased compared to Newtonian mechanics, from 3N to n = 3N − C coupled second-order differential equations in the generalized coordinates. These equations do not include constraint forces at all, only non-constraint forces need to be accounted for. Although the equations of motion include <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a>, the results of the partial derivatives are still <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> in the position coordinates of the particles. The <a href="/wiki/Total_derivative" title="Total derivative">total time derivative</a> denoted d/dt often involves <a href="/wiki/Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">implicit differentiation</a>. Both equations are linear in the Lagrangian, but generally are nonlinear coupled equations in the coordinates. <div class="mw-heading mw-heading2"><h2 id="From_Newtonian_to_Lagrangian_mechanics">From Newtonian to Lagrangian mechanics</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=3" title="Edit section: From Newtonian to Lagrangian mechanics">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Newton's_laws">Newton's laws</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=4" title="Edit section: Newton's laws">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:GodfreyKneller-IsaacNewton-1689.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/150px-GodfreyKneller-IsaacNewton-1689.jpg" decoding="async" width="150" height="211" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/225px-GodfreyKneller-IsaacNewton-1689.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/300px-GodfreyKneller-IsaacNewton-1689.jpg 2x" data-file-width="1364" data-file-height="1916" /></a><figcaption><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> (1642–1727)</figcaption></figure> For simplicity, Newton's laws can be illustrated for one particle without much loss of generality (for a system of N particles, all of these equations apply to each particle in the system). The <a href="/wiki/Equation_of_motion" class="mw-redirect" title="Equation of motion">equation of motion</a> for a particle of constant mass m is <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a> of 1687, in modern vector notation <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} ,}"> where a is its acceleration and F the resultant force acting on it. Where the mass is varying, the equation needs to be generalised to take the time derivative of the momentum. In three spatial dimensions, this is a system of three coupled second-order <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> to solve, since there are three components in this vector equation. The solution is the position vector r of the particle at time t, subject to the <a href="/wiki/Initial_condition" title="Initial condition">initial conditions</a> of r and v when t = 0. Newton's laws are easy to use in Cartesian coordinates, but Cartesian coordinates are not always convenient, and for other coordinate systems the equations of motion can become complicated. In a set of <a href="/wiki/Curvilinear_coordinates" title="Curvilinear coordinates">curvilinear coordinates</a> ξ = (ξ1, ξ2, ξ3), the law in <a href="/wiki/Tensor_index_notation" class="mw-redirect" title="Tensor index notation">tensor index notation</a> is the "Lagrangian form"<a href="#cite_note-Kay_1988-19">[18]</a><a href="#cite_note-Synge_1949-20">[19]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{a}=m\left({\frac {\mathrm {d} ^{2}\xi ^{a}}{\mathrm {d} t^{2}}}+\Gamma ^{a}{}_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}\right)=g^{ak}\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\xi }}^{k}}}-{\frac {\partial T}{\partial \xi ^{k}}}\right),\quad {\dot {\xi }}^{a}\equiv {\frac {\mathrm {d} \xi ^{a}}{\mathrm {d} t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>k</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{a}=m\left({\frac {\mathrm {d} ^{2}\xi ^{a}}{\mathrm {d} t^{2}}}+\Gamma ^{a}{}_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}\right)=g^{ak}\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\xi }}^{k}}}-{\frac {\partial T}{\partial \xi ^{k}}}\right),\quad {\dot {\xi }}^{a}\equiv {\frac {\mathrm {d} \xi ^{a}}{\mathrm {d} t}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de9ef06d7225735f3e9d1b97cf3ba8cfbda4fc22" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:71.976ex; height:7.843ex;" alt="{\displaystyle F^{a}=m\left({\frac {\mathrm {d} ^{2}\xi ^{a}}{\mathrm {d} t^{2}}}+\Gamma ^{a}{}_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}\right)=g^{ak}\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\xi }}^{k}}}-{\frac {\partial T}{\partial \xi ^{k}}}\right),\quad {\dot {\xi }}^{a}\equiv {\frac {\mathrm {d} \xi ^{a}}{\mathrm {d} t}},}"> where Fa is the a-th <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant component</a> of the resultant force acting on the particle, Γabc are the <a href="/wiki/Christoffel_symbols" title="Christoffel symbols">Christoffel symbols</a> of the second kind, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {1}{2}}mg_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {1}{2}}mg_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42c7e6a2757ac503662cd362cd745c7e33bec793" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.739ex; height:6.009ex;" alt="{\displaystyle T={\frac {1}{2}}mg_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}}"> is the kinetic energy of the particle, and gbc the <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariant components</a> of the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> of the curvilinear coordinate system. All the indices a, b, c, each take the values 1, 2, 3. Curvilinear coordinates are not the same as generalized coordinates. It may seem like an overcomplication to cast Newton's law in this form, but there are advantages. The acceleration components in terms of the Christoffel symbols can be avoided by evaluating derivatives of the kinetic energy instead. If there is no resultant force acting on the particle, F = 0, it does not accelerate, but moves with constant velocity in a straight line. Mathematically, the solutions of the differential equation are <a href="/wiki/Geodesic" title="Geodesic">geodesics</a>, the curves of extremal length between two points in space (these may end up being minimal, that is the shortest paths, but not necessarily). In flat 3D real space the geodesics are simply straight lines. So for a free particle, Newton's second law coincides with the geodesic equation and states that free particles follow geodesics, the extremal trajectories it can move along. If the particle is subject to forces F ≠ 0, the particle accelerates due to forces acting on it and deviates away from the geodesics it would follow if free. With appropriate extensions of the quantities given here in flat 3D space to 4D <a href="/wiki/Curved_spacetime" title="Curved spacetime">curved spacetime</a>, the above form of Newton's law also carries over to <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a>'s <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, in which case free particles follow geodesics in curved spacetime that are no longer "straight lines" in the ordinary sense.<a href="#cite_note-21">[20]</a> However, we still need to know the total resultant force F acting on the particle, which in turn requires the resultant non-constraint force N plus the resultant constraint force C, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\mathbf {C} +\mathbf {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\mathbf {C} +\mathbf {N} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62751bcfb0add5e5f85ec3bee9defb0738a520dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.291ex; height:2.343ex;" alt="{\displaystyle \mathbf {F} =\mathbf {C} +\mathbf {N} .}"> The constraint forces can be complicated, since they generally depend on time. Also, if there are constraints, the curvilinear coordinates are not independent but related by one or more constraint equations. The constraint forces can either be eliminated from the equations of motion, so only the non-constraint forces remain, or included by including the constraint equations in the equations of motion. <div class="mw-heading mw-heading3"><h3 id="D'Alembert's_principle">D'Alembert's principle</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=5" title="Edit section: D'Alembert's principle">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Alembert.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Alembert.jpg/150px-Alembert.jpg" decoding="async" width="150" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Alembert.jpg/225px-Alembert.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Alembert.jpg/300px-Alembert.jpg 2x" data-file-width="1024" data-file-height="1280" /></a><figcaption><a href="/wiki/Jean_d%27Alembert" class="mw-redirect" title="Jean d'Alembert">Jean d'Alembert</a> (1717—1783)</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:408px;max-width:408px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><a href="/wiki/File:Constraint_force_virtual_displacement_1_dof.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Constraint_force_virtual_displacement_1_dof.svg/200px-Constraint_force_virtual_displacement_1_dof.svg.png" decoding="async" width="200" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Constraint_force_virtual_displacement_1_dof.svg/300px-Constraint_force_virtual_displacement_1_dof.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Constraint_force_virtual_displacement_1_dof.svg/400px-Constraint_force_virtual_displacement_1_dof.svg.png 2x" data-file-width="354" data-file-height="289" /></a></div><div class="thumbcaption">One degree of freedom.</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><a href="/wiki/File:Constraint_force_virtual_displacement_2_dof.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Constraint_force_virtual_displacement_2_dof.svg/200px-Constraint_force_virtual_displacement_2_dof.svg.png" decoding="async" width="200" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Constraint_force_virtual_displacement_2_dof.svg/300px-Constraint_force_virtual_displacement_2_dof.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Constraint_force_virtual_displacement_2_dof.svg/400px-Constraint_force_virtual_displacement_2_dof.svg.png 2x" data-file-width="336" data-file-height="292" /></a></div><div class="thumbcaption">Two degrees of freedom.</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Constraint force C and virtual displacement δr for a particle of mass m confined to a curve. The resultant non-constraint force is N.</div></div></div></div> A fundamental result in <a href="/wiki/Analytical_mechanics" title="Analytical mechanics">analytical mechanics</a> is <a href="/wiki/D%27Alembert%27s_principle" title="D'Alembert's principle">D'Alembert's principle</a>, introduced in 1708 by <a href="/wiki/Jacques_Bernoulli" class="mw-redirect" title="Jacques Bernoulli">Jacques Bernoulli</a> to understand <a href="/wiki/Static_equilibrium" class="mw-redirect" title="Static equilibrium">static equilibrium</a>, and developed by <a href="/wiki/D%27Alembert" class="mw-redirect" title="D'Alembert">D'Alembert</a> in 1743 to solve dynamical problems.<a href="#cite_note-22">[21]</a> The principle asserts for N particles the virtual work, i.e. the work along a virtual displacement, δrk, is zero:<a href="#cite_note-Torby_1984_page=269-9">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}+\mathbf {C} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}+\mathbf {C} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/345ccc6071ed4024c5c361effc15a62d0908ee90" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.389ex; height:7.343ex;" alt="{\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}+\mathbf {C} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.}"> The <a href="/wiki/Virtual_displacement" title="Virtual displacement">virtual displacements</a>, δrk, are by definition infinitesimal changes in the configuration of the system consistent with the constraint forces acting on the system at an instant of time,<a href="#cite_note-Goldstein_1980_page=16–18-23">[22]</a> i.e. in such a way that the constraint forces maintain the constrained motion. They are not the same as the actual displacements in the system, which are caused by the resultant constraint and non-constraint forces acting on the particle to accelerate and move it.<a href="#cite_note-24">[nb 2]</a> <a href="/wiki/Virtual_work" title="Virtual work">Virtual work</a> is the work done along a virtual displacement for any force (constraint or non-constraint). Since the constraint forces act perpendicular to the motion of each particle in the system to maintain the constraints, the total virtual work by the constraint forces acting on the system is zero:<a href="#cite_note-25">[23]</a><a href="#cite_note-26">[nb 3]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{N}\mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{N}\mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e8e186f3d790c9fb21731b4950d4f173d4bf19b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.588ex; height:7.343ex;" alt="{\displaystyle \sum _{k=1}^{N}\mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0,}"> so that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d237bc583e120c910f729d8fab7c58d5fe4e829" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.529ex; height:7.343ex;" alt="{\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.}"> Thus D'Alembert's principle allows us to concentrate on only the applied non-constraint forces, and exclude the constraint forces in the equations of motion.<a href="#cite_note-27">[24]</a><a href="#cite_note-28">[25]</a> The form shown is also independent of the choice of coordinates. However, it cannot be readily used to set up the equations of motion in an arbitrary coordinate system since the displacements δrk might be connected by a constraint equation, which prevents us from setting the N individual summands to 0. We will therefore seek a system of mutually independent coordinates for which the total sum will be 0 if and only if the individual summands are 0. Setting each of the summands to 0 will eventually give us our separated equations of motion. <div class="mw-heading mw-heading3"><h3 id="Equations_of_motion_from_D'Alembert's_principle">Equations of motion from D'Alembert's principle</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=6" title="Edit section: Equations of motion from D'Alembert's principle">edit</a>]</div> If there are constraints on particle k, then since the coordinates of the position rk = (xk, yk, zk) are linked together by a constraint equation, so are those of the <a href="/wiki/Virtual_displacement" title="Virtual displacement">virtual displacements</a> δrk = (δxk, δyk, δzk). Since the generalized coordinates are independent, we can avoid the complications with the δrk by converting to virtual displacements in the generalized coordinates. These are related in the same form as a <a href="/wiki/Total_differential" class="mw-redirect" title="Total differential">total differential</a>,<a href="#cite_note-Torby_1984_page=269-9">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \mathbf {r} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \mathbf {r} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29058e05bfcaf903a00c92e298b572e50b99114a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.067ex; height:7.176ex;" alt="{\displaystyle \delta \mathbf {r} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}.}"> There is no partial time derivative with respect to time multiplied by a time increment, since this is a virtual displacement, one along the constraints in an instant of time. The first term in D'Alembert's principle above is the virtual work done by the non-constraint forces Nk along the virtual displacements δrk, and can without loss of generality be converted into the generalized analogues by the definition of <a href="/wiki/Generalized_force" class="mw-redirect" title="Generalized force">generalized forces</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{j}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{j}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3de913cee5de1b2841b31dc5b79859225c28b96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.44ex; height:7.343ex;" alt="{\displaystyle Q_{j}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}},}"> so that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{N}\mathbf {N} _{k}\cdot \delta \mathbf {r} _{k}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot \sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}=\sum _{j=1}^{n}Q_{j}\delta q_{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{N}\mathbf {N} _{k}\cdot \delta \mathbf {r} _{k}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot \sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}=\sum _{j=1}^{n}Q_{j}\delta q_{j}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87109ba3ec144be2632dc6f220861c90a9f0639c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:47.854ex; height:7.676ex;" alt="{\displaystyle \sum _{k=1}^{N}\mathbf {N} _{k}\cdot \delta \mathbf {r} _{k}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot \sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}=\sum _{j=1}^{n}Q_{j}\delta q_{j}.}"> This is half of the conversion to generalized coordinates. It remains to convert the acceleration term into generalized coordinates, which is not immediately obvious. Recalling the Lagrange form of Newton's second law, the partial derivatives of the kinetic energy with respect to the generalized coordinates and velocities can be found to give the desired result:<a href="#cite_note-Torby_1984_page=269-9">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d19530de0ccb787d5b740987205b23d08e63189b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.379ex; height:7.343ex;" alt="{\displaystyle \sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}.}"> Now D'Alembert's principle is in the generalized coordinates as required, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=1}^{n}\left[Q_{j}-\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}\right)\right]\delta q_{j}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=1}^{n}\left[Q_{j}-\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}\right)\right]\delta q_{j}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aabd545f28d06dcc1b1f7bef3f89519cc2194c0d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.363ex; height:7.676ex;" alt="{\displaystyle \sum _{j=1}^{n}\left[Q_{j}-\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}\right)\right]\delta q_{j}=0,}"> and since these virtual displacements δqj are independent and nonzero, the coefficients can be equated to zero, resulting in Lagrange's equations<a href="#cite_note-29">[26]</a><a href="#cite_note-30">[27]</a> or the generalized equations of motion,<a href="#cite_note-31">[28]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d680e28c77d3fd077ef7dfbd3d06376aabc823e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.197ex; height:6.343ex;" alt="{\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}}"> These equations are equivalent to Newton's laws for the non-constraint forces. The generalized forces in this equation are derived from the non-constraint forces only – the constraint forces have been excluded from D'Alembert's principle and do not need to be found. The generalized forces may be non-conservative, provided they satisfy D'Alembert's principle.<a href="#cite_note-32">[29]</a> <div class="mw-heading mw-heading3"><h3 id="Euler–Lagrange_equations_and_Hamilton's_principle">Euler–Lagrange equations and Hamilton's principle</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=7" title="Edit section: Euler–Lagrange equations and Hamilton's principle">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Least_action_principle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Least_action_principle.svg/250px-Least_action_principle.svg.png" decoding="async" width="250" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Least_action_principle.svg/375px-Least_action_principle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Least_action_principle.svg/500px-Least_action_principle.svg.png 2x" data-file-width="318" data-file-height="232" /></a><figcaption>As the system evolves, q traces a path through <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a> (only some are shown). The path taken by the system (red) has a stationary action (δS = 0) under small changes in the configuration of the system (δq).<a href="#cite_note-penrose-33">[30]</a></figcaption></figure> For a non-conservative force which depends on velocity, it may be possible to find a potential energy function V that depends on positions and velocities. If the generalized forces Qi can be derived from a potential V such that<a href="#cite_note-34">[31]</a><a href="#cite_note-35">[32]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial V}{\partial {\dot {q}}_{j}}}-{\frac {\partial V}{\partial q_{j}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </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>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial V}{\partial {\dot {q}}_{j}}}-{\frac {\partial V}{\partial q_{j}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4be508163d6beb9c8b8ad8624e91790c0904bac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.844ex; height:6.343ex;" alt="{\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial V}{\partial {\dot {q}}_{j}}}-{\frac {\partial V}{\partial q_{j}}},}"> equating to Lagrange's equations and defining the Lagrangian as L = T − V obtains Lagrange's equations of the second kind or the Euler–Lagrange equations of motion <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb947b09ec5029f6c8bec3a1cf6f2c8a3214d808" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.258ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}=0.}"> However, the Euler–Lagrange equations can only account for non-conservative forces if a potential can be found as shown. This may not always be possible for non-conservative forces, and Lagrange's equations do not involve any potential, only generalized forces; therefore they are more general than the Euler–Lagrange equations. The Euler–Lagrange equations also follow from the <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a>. The variation of the Lagrangian is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta L=\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta {\dot {q}}_{j}\right),\quad \delta {\dot {q}}_{j}\equiv \delta {\frac {\mathrm {d} q_{j}}{\mathrm {d} t}}\equiv {\frac {\mathrm {d} (\delta q_{j})}{\mathrm {d} t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>L</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>≡</mo> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta L=\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta {\dot {q}}_{j}\right),\quad \delta {\dot {q}}_{j}\equiv \delta {\frac {\mathrm {d} q_{j}}{\mathrm {d} t}}\equiv {\frac {\mathrm {d} (\delta q_{j})}{\mathrm {d} t}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45ea23e02ece639644e951e173323fb887940844" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:56.847ex; height:7.676ex;" alt="{\displaystyle \delta L=\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta {\dot {q}}_{j}\right),\quad \delta {\dot {q}}_{j}\equiv \delta {\frac {\mathrm {d} q_{j}}{\mathrm {d} t}}\equiv {\frac {\mathrm {d} (\delta q_{j})}{\mathrm {d} t}},}"> which has a form similar to the <a href="/wiki/Total_differential" class="mw-redirect" title="Total differential">total differential</a> of L, but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements. An <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a> with respect to time can transfer the time derivative of δqj to the ∂L/∂(dqj/dt), in the process exchanging d(δqj)/dt for δqj, allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t&=\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)\,\mathrm {d} t\\&=\sum _{j=1}^{n}\left[{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right]_{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)\delta q_{j}\,\mathrm {d} t.\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> <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>δ</mi> <mi>L</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mtd> <mtd> <mi></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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>]</mo> </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> <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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t&=\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)\,\mathrm {d} t\\&=\sum _{j=1}^{n}\left[{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right]_{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)\delta q_{j}\,\mathrm {d} t.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739a857b0aa09f6fc8bbe13cdd53e01376d9bba4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:66.003ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}\int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t&=\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)\,\mathrm {d} t\\&=\sum _{j=1}^{n}\left[{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right]_{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)\delta q_{j}\,\mathrm {d} t.\end{aligned}}}"> Now, if the condition δqj(t1) = δqj(t2) = 0 holds for all j, the terms not integrated are zero. If in addition the entire time integral of δL is zero, then because the δqj are independent, and the only way for a definite integral to be zero is if the integrand equals zero, each of the coefficients of δqj must also be zero. Then we obtain the equations of motion. This can be summarized by <a href="/wiki/Hamilton%27s_principle" title="Hamilton's principle">Hamilton's principle</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>δ</mi> <mi>L</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6991ae3649644a106b6c23daef4711df46be1d14" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.567ex; height:6.509ex;" alt="{\displaystyle \int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t=0.}"> The time integral of the Lagrangian is another quantity called the <a href="/wiki/Action_(physics)" title="Action (physics)">action</a>, defined as<a href="#cite_note-36">[33]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\int _{t_{1}}^{t_{2}}L\,\mathrm {d} t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>L</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\int _{t_{1}}^{t_{2}}L\,\mathrm {d} t,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7541259ce51ac84a029d8a5bb2a8ffbc7c58796d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.855ex; height:6.509ex;" alt="{\displaystyle S=\int _{t_{1}}^{t_{2}}L\,\mathrm {d} t,}"> which is a <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functional</a>; it takes in the Lagrangian function for all times between t1 and t2 and returns a scalar value. Its dimensions are the same as [<a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>], [energy]·[time], or [length]·[momentum]. With this definition Hamilton's principle is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta S=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>S</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta S=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be02bab96c7b908ef7d379d67f1d3aca2fdfde29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.456ex; height:2.343ex;" alt="{\displaystyle \delta S=0.}"> Instead of thinking about particles accelerating in response to applied forces, one might think of them picking out the path with a stationary action, with the end points of the path in configuration space held fixed at the initial and final times. Hamilton's principle is one of several <a href="/wiki/Action_principles" title="Action principles">action principles</a>.<a href="#cite_note-HancTaylorTuleja-37">[34]</a> Historically, the idea of finding the shortest path a particle can follow subject to a force motivated the first applications of the <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a> to mechanical problems, such as the <a href="/wiki/Brachistochrone_problem" class="mw-redirect" title="Brachistochrone problem">Brachistochrone problem</a> solved by <a href="/wiki/Jean_Bernoulli" class="mw-redirect" title="Jean Bernoulli">Jean Bernoulli</a> in 1696, as well as <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a>, <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a>, <a href="/wiki/Guillaume_de_l%27H%C3%B4pital" title="Guillaume de l'Hôpital">L'Hôpital</a> around the same time, and <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> the following year.<a href="#cite_note-Hand_1998_page=44–45-38">[35]</a> Newton himself was thinking along the lines of the variational calculus, but did not publish.<a href="#cite_note-Hand_1998_page=44–45-38">[35]</a> These ideas in turn lead to the <a href="/wiki/Variational_principle" title="Variational principle">variational principles</a> of mechanics, of <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Fermat</a>, <a href="/wiki/Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a>, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a>, <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a>, and others. Hamilton's principle can be applied to <a href="/wiki/Nonholonomic_constraints" class="mw-redirect" title="Nonholonomic constraints">nonholonomic constraints</a> if the constraint equations can be put into a certain form, a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of first order differentials in the coordinates. The resulting constraint equation can be rearranged into first order differential equation.<a href="#cite_note-39">[36]</a> This will not be given here. <div class="mw-heading mw-heading3"><h3 id="Lagrange_multipliers_and_constraints">Lagrange multipliers and constraints</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=8" title="Edit section: Lagrange multipliers and constraints">edit</a>]</div> The Lagrangian L can be varied in the Cartesian rk coordinates, for N particles, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee97d395e9f68bd1f378be6841d7623f1240cd2a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.577ex; height:7.343ex;" alt="{\displaystyle \int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0.}"> Hamilton's principle is still valid even if the coordinates L is expressed in are not independent, here rk, but the constraints are still assumed to be holonomic.<a href="#cite_note-40">[37]</a> As always the end points are fixed δrk(t1) = δrk(t2) = 0 for all k. What cannot be done is to simply equate the coefficients of δrk to zero because the δrk are not independent. Instead, the method of <a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multipliers</a> can be used to include the constraints. Multiplying each constraint equation fi(rk, t) = 0 by a Lagrange multiplier λi for i = 1, 2, ..., C, and adding the results to the original Lagrangian, gives the new Lagrangian <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L'=L(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,{\dot {\mathbf {r} }}_{1},{\dot {\mathbf {r} }}_{2},\ldots ,t)+\sum _{i=1}^{C}\lambda _{i}(t)f_{i}(\mathbf {r} _{k},t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L'=L(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,{\dot {\mathbf {r} }}_{1},{\dot {\mathbf {r} }}_{2},\ldots ,t)+\sum _{i=1}^{C}\lambda _{i}(t)f_{i}(\mathbf {r} _{k},t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0143c86f1de8500648a9f3d020ae235e5552fb7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.614ex; height:7.343ex;" alt="{\displaystyle L'=L(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,{\dot {\mathbf {r} }}_{1},{\dot {\mathbf {r} }}_{2},\ldots ,t)+\sum _{i=1}^{C}\lambda _{i}(t)f_{i}(\mathbf {r} _{k},t).}"> The Lagrange multipliers are arbitrary functions of time t, but not functions of the coordinates rk, so the multipliers are on equal footing with the position coordinates. Varying this new Lagrangian and integrating with respect to time gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{t_{1}}^{t_{2}}\delta L'\mathrm {d} t=\int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>δ</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{t_{1}}^{t_{2}}\delta L'\mathrm {d} t=\int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a628785d32892bdd3b4277aa1c9ce9e580b68faa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.975ex; height:7.509ex;" alt="{\displaystyle \int _{t_{1}}^{t_{2}}\delta L'\mathrm {d} t=\int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0.}"> The introduced multipliers can be found so that the coefficients of δrk are zero, even though the rk are not independent. The equations of motion follow. From the preceding analysis, obtaining the solution to this integral is equivalent to the statement <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial L'}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\mathbf {r} }}_{k}}}=0\quad \Rightarrow \quad {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial L'}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\mathbf {r} }}_{k}}}=0\quad \Rightarrow \quad {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd24efbab636e94c93f914b80644428f4e71e5e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:59.722ex; height:7.343ex;" alt="{\displaystyle {\frac {\partial L'}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\mathbf {r} }}_{k}}}=0\quad \Rightarrow \quad {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}"> which are Lagrange's equations of the first kind. Also, the λi Euler-Lagrange equations for the new Lagrangian return the constraint equations <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial L'}{\partial \lambda _{i}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\lambda }}_{i}}}=0\quad \Rightarrow \quad f_{i}(\mathbf {r} _{k},t)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial L'}{\partial \lambda _{i}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\lambda }}_{i}}}=0\quad \Rightarrow \quad f_{i}(\mathbf {r} _{k},t)=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9b132ad2ec5757f510468c9206c52ddefae41d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:39.893ex; height:6.509ex;" alt="{\displaystyle {\frac {\partial L'}{\partial \lambda _{i}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\lambda }}_{i}}}=0\quad \Rightarrow \quad f_{i}(\mathbf {r} _{k},t)=0.}"> For the case of a conservative force given by the gradient of some potential energy V, a function of the rk coordinates only, substituting the Lagrangian L = T − V gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \underbrace {{\frac {\partial T}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\mathbf {r} }}_{k}}}} _{-\mathbf {F} _{k}}+\underbrace {-{\frac {\partial V}{\partial \mathbf {r} _{k}}}} _{\mathbf {N} _{k}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </munder> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <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"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </munder> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {{\frac {\partial T}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\mathbf {r} }}_{k}}}} _{-\mathbf {F} _{k}}+\underbrace {-{\frac {\partial V}{\partial \mathbf {r} _{k}}}} _{\mathbf {N} _{k}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d826bb0cb54d1fe1ea8a62df3372883787565352" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:41.543ex; height:10.509ex;" alt="{\displaystyle \underbrace {{\frac {\partial T}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\mathbf {r} }}_{k}}}} _{-\mathbf {F} _{k}}+\underbrace {-{\frac {\partial V}{\partial \mathbf {r} _{k}}}} _{\mathbf {N} _{k}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}"> and identifying the derivatives of kinetic energy as the (negative of the) resultant force, and the derivatives of the potential equaling the non-constraint force, it follows the constraint forces are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} _{k}=\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} _{k}=\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad92b79b7e73aed44e4e7336848d3277b09a8bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.007ex; height:7.343ex;" alt="{\displaystyle \mathbf {C} _{k}=\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}},}"> thus giving the constraint forces explicitly in terms of the constraint equations and the Lagrange multipliers. <div class="mw-heading mw-heading2"><h2 id="Properties_of_the_Lagrangian">Properties of the Lagrangian</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=9" title="Edit section: Properties of the Lagrangian">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Non-uniqueness">Non-uniqueness</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=10" title="Edit section: Non-uniqueness">edit</a>]</div> The Lagrangian of a given system is not unique. A Lagrangian L can be multiplied by a nonzero constant a and shifted by an arbitrary constant b, and the new Lagrangian L′ = aL + b will describe the same motion as L. If one restricts as above to trajectories q over a given time interval [tst, tfin]} and fixed end points Pst = q(tst) and Pfin = q(tfin), then two Lagrangians describing the same system can differ by the "total time derivative" of a function f(q, t):<a href="#cite_note-Landau_1976_page=4-41">[38]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L'(\mathbf {q} ,{\dot {\mathbf {q} }},t)=L(\mathbf {q} ,{\dot {\mathbf {q} }},t)+{\frac {\mathrm {d} f(\mathbf {q} ,t)}{\mathrm {d} t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L'(\mathbf {q} ,{\dot {\mathbf {q} }},t)=L(\mathbf {q} ,{\dot {\mathbf {q} }},t)+{\frac {\mathrm {d} f(\mathbf {q} ,t)}{\mathrm {d} t}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e96c74cdf6cfc855130378988ec3b8330420154" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.038ex; height:5.843ex;" alt="{\displaystyle L'(\mathbf {q} ,{\dot {\mathbf {q} }},t)=L(\mathbf {q} ,{\dot {\mathbf {q} }},t)+{\frac {\mathrm {d} f(\mathbf {q} ,t)}{\mathrm {d} t}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\mathrm {d} f(\mathbf {q} ,t)}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\mathrm {d} f(\mathbf {q} ,t)}{\mathrm {d} t}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953e306736122120b417dea17575b65199bf3f7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.986ex; height:4.343ex;" alt="{\textstyle {\frac {\mathrm {d} f(\mathbf {q} ,t)}{\mathrm {d} t}}}"> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\partial f(\mathbf {q} ,t)}{\partial t}}+\sum _{i}{\frac {\partial f(\mathbf {q} ,t)}{\partial q_{i}}}{\dot {q}}_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\partial f(\mathbf {q} ,t)}{\partial t}}+\sum _{i}{\frac {\partial f(\mathbf {q} ,t)}{\partial q_{i}}}{\dot {q}}_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6555ab62d7a8a310ea0670fc2c9b6bf917c59947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.312ex; height:4.676ex;" alt="{\textstyle {\frac {\partial f(\mathbf {q} ,t)}{\partial t}}+\sum _{i}{\frac {\partial f(\mathbf {q} ,t)}{\partial q_{i}}}{\dot {q}}_{i}.}"> Both Lagrangians L and L′ produce the same equations of motion<a href="#cite_note-42">[39]</a><a href="#cite_note-43">[40]</a> since the corresponding actions S and S′ are related via <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S'[\mathbf {q} ]&=\int _{t_{\text{st}}}^{t_{\text{fin}}}L'(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt\\&=\int _{t_{\text{st}}}^{t_{\text{fin}}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt+\int _{t_{\text{st}}}^{t_{\text{fin}}}{\frac {\mathrm {d} f(\mathbf {q} (t),t)}{\mathrm {d} t}}\,dt\\&=S[\mathbf {q} ]+f(P_{\text{fin}},t_{\text{fin}})-f(P_{\text{st}},t_{\text{st}}),\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> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>st</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fin</mtext> </mrow> </msub> </mrow> </msubsup> <msup> <mi>L</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>st</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fin</mtext> </mrow> </msub> </mrow> </msubsup> <mi>L</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>st</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fin</mtext> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">]</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fin</mtext> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fin</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>st</mtext> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>st</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S'[\mathbf {q} ]&=\int _{t_{\text{st}}}^{t_{\text{fin}}}L'(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt\\&=\int _{t_{\text{st}}}^{t_{\text{fin}}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt+\int _{t_{\text{st}}}^{t_{\text{fin}}}{\frac {\mathrm {d} f(\mathbf {q} (t),t)}{\mathrm {d} t}}\,dt\\&=S[\mathbf {q} ]+f(P_{\text{fin}},t_{\text{fin}})-f(P_{\text{st}},t_{\text{st}}),\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b08287b06286f1d5e8f24c6e1ac4d084903a78" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:52.801ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}S'[\mathbf {q} ]&=\int _{t_{\text{st}}}^{t_{\text{fin}}}L'(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt\\&=\int _{t_{\text{st}}}^{t_{\text{fin}}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt+\int _{t_{\text{st}}}^{t_{\text{fin}}}{\frac {\mathrm {d} f(\mathbf {q} (t),t)}{\mathrm {d} t}}\,dt\\&=S[\mathbf {q} ]+f(P_{\text{fin}},t_{\text{fin}})-f(P_{\text{st}},t_{\text{st}}),\end{aligned}}}"> with the last two components f(Pfin, tfin) and f(Pst, tst) independent of q. <div class="mw-heading mw-heading3"><h3 id="Invariance_under_point_transformations">Invariance under point transformations</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=11" title="Edit section: Invariance under point transformations">edit</a>]</div> Given a set of generalized coordinates q, if we change these variables to a new set of generalized coordinates Q according to a <a href="/wiki/Canonical_transformation" title="Canonical transformation">point transformation</a> Q = Q(q, t) which is invertible as q = q(Q, t), the new Lagrangian L′ is a function of the new coordinates <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L'(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=L(\mathbf {q} (\mathbf {Q} ,t),{\dot {\mathbf {q} }}(\mathbf {Q} ,{\dot {\mathbf {Q} }},t),t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L'(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=L(\mathbf {q} (\mathbf {Q} ,t),{\dot {\mathbf {q} }}(\mathbf {Q} ,{\dot {\mathbf {Q} }},t),t),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d336eb316aa857c7bf2eec29d66d7c55b071e87" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.3ex; height:3.176ex;" alt="{\displaystyle L'(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=L(\mathbf {q} (\mathbf {Q} ,t),{\dot {\mathbf {q} }}(\mathbf {Q} ,{\dot {\mathbf {Q} }},t),t),}"> and by the <a href="/wiki/Chain_rule#Higher_dimensions" title="Chain rule">chain rule</a> for partial differentiation, Lagrange's equations are invariant under this transformation;<a href="#cite_note-44">[41]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {Q}}_{i}}}={\frac {\partial L'}{\partial Q_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {Q}}_{i}}}={\frac {\partial L'}{\partial Q_{i}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5e75863c49c32cfe31a5ee2cc7e408d2bd2228" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.298ex; height:6.676ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {Q}}_{i}}}={\frac {\partial L'}{\partial Q_{i}}}.}"> This may simplify the equations of motion. <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style="">Proof For a coordinate transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{i}=Q_{i}(\mathbf {q} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{i}=Q_{i}(\mathbf {q} ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb4111a3e8165e3166d6bc2b35edf6d9151a8d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.473ex; height:2.843ex;" alt="{\displaystyle Q_{i}=Q_{i}(\mathbf {q} ,t)}">, we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d{Q_{i}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}d{q_{k}}+{\frac {\partial Q_{k}}{\partial t}}dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d{Q_{i}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}d{q_{k}}+{\frac {\partial Q_{k}}{\partial t}}dt,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54688dd31ce661b419e4d545e9b32c4a762d189f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.452ex; height:6.509ex;" alt="{\displaystyle d{Q_{i}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}d{q_{k}}+{\frac {\partial Q_{k}}{\partial t}}dt,}"> which implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q_{i}}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial Q_{k}}{\partial t}}(\mathbf {q} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q_{i}}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial Q_{k}}{\partial t}}(\mathbf {q} ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1321689efa8df15035bd566eaf75febf98a85701" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.242ex; height:6.509ex;" alt="{\displaystyle {\dot {Q_{i}}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial Q_{k}}{\partial t}}(\mathbf {q} ,t)}"> which implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {\dot {Q_{i}}}}{\partial {\dot {q}}_{k}}}={\frac {\partial Q_{i}}{\partial q_{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\dot {Q_{i}}}}{\partial {\dot {q}}_{k}}}={\frac {\partial Q_{i}}{\partial q_{k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9e2aa2bc99006b5729f477dfde75bf95a286184" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.683ex; height:6.509ex;" alt="{\displaystyle {\frac {\partial {\dot {Q_{i}}}}{\partial {\dot {q}}_{k}}}={\frac {\partial Q_{i}}{\partial q_{k}}}}">. It also follows that: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {\dot {Q_{i}}}}{\partial q_{j}}}=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{j}\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial q_{j}\partial t}}(\mathbf {q} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\dot {Q_{i}}}}{\partial q_{j}}}=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{j}\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial q_{j}\partial t}}(\mathbf {q} ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37770229c58402fb64f41e39f65351455c6d8e23" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.326ex; height:7.009ex;" alt="{\displaystyle {\frac {\partial {\dot {Q_{i}}}}{\partial q_{j}}}=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{j}\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial q_{j}\partial t}}(\mathbf {q} ,t)}"> and similarly: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}\left({\frac {\partial {Q_{i}}}{\partial q_{j}}}\right)=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{k}\partial q_{j}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial t\partial q_{j}}}(\mathbf {q} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\left({\frac {\partial {Q_{i}}}{\partial q_{j}}}\right)=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{k}\partial q_{j}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial t\partial q_{j}}}(\mathbf {q} ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c192324bc899432a0b700425a70d9922cbf4a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.026ex; height:6.843ex;" alt="{\displaystyle {\frac {d}{dt}}\left({\frac {\partial {Q_{i}}}{\partial q_{j}}}\right)=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{k}\partial q_{j}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial t\partial q_{j}}}(\mathbf {q} ,t)}"> which imply that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}\left({\frac {\partial Q_{i}}{\partial q_{k}}}\right)={\frac {\partial {\dot {Q}}_{i}}{\partial q_{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\left({\frac {\partial Q_{i}}{\partial q_{k}}}\right)={\frac {\partial {\dot {Q}}_{i}}{\partial q_{k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba97649312ae7cec59f5523bdb4ebf23e4094a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.383ex; height:6.676ex;" alt="{\displaystyle {\frac {d}{dt}}\left({\frac {\partial Q_{i}}{\partial q_{k}}}\right)={\frac {\partial {\dot {Q}}_{i}}{\partial q_{k}}}}">. The two derived relations can be employed in the proof. Starting from Euler Lagrange equations in initial set of generalized coordinates, we have: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial q_{i}}}=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial Q_{k}}{\partial {q}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {q}_{i}}}\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right){\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}+{\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial Q_{k}}{\partial q_{i}}}\right)\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right)-{\frac {\partial L}{\partial Q_{k}}}\right){\frac {\partial Q_{k}}{\partial q_{i}}}=0\\\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> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <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"> <mi mathvariant="normal">∂</mi> </mrow> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> </mrow> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <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"> <mi mathvariant="normal">∂</mi> </mrow> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <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> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> </mrow> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <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"> <mi mathvariant="normal">∂</mi> </mrow> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <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> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <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> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial q_{i}}}=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial Q_{k}}{\partial {q}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {q}_{i}}}\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right){\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}+{\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial Q_{k}}{\partial q_{i}}}\right)\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right)-{\frac {\partial L}{\partial Q_{k}}}\right){\frac {\partial Q_{k}}{\partial q_{i}}}=0\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce102948237eed8ad533768477bfbe88b9e5ee7e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.005ex; width:82.614ex; height:29.009ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial q_{i}}}=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial Q_{k}}{\partial {q}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {q}_{i}}}\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right){\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}+{\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial Q_{k}}{\partial q_{i}}}\right)\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right)-{\frac {\partial L}{\partial Q_{k}}}\right){\frac {\partial Q_{k}}{\partial q_{i}}}=0\\\end{aligned}}}"> Since the transformation from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\rightarrow Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcaa7f58f67e03c478113ad4d10fd9ac53a87401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.522ex; height:2.509ex;" alt="{\displaystyle q\rightarrow Q}"> is invertible, it follows that the form of the Euler-Lagrange equation is invariant i.e., <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {Q}}_{i}}}-{\frac {{\partial }L}{\partial Q_{i}}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <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"> <mi mathvariant="normal">∂</mi> </mrow> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {Q}}_{i}}}-{\frac {{\partial }L}{\partial Q_{i}}}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef565493466b2f0fae2ec58b9113d96082a930b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.224ex; height:6.509ex;" alt="{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {Q}}_{i}}}-{\frac {{\partial }L}{\partial Q_{i}}}=0.}"> </div> <div class="mw-heading mw-heading3"><h3 id="Cyclic_coordinates_and_conserved_momenta">Cyclic coordinates and conserved momenta</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=12" title="Edit section: Cyclic coordinates and conserved momenta">edit</a>]</div> An important property of the Lagrangian is that <a href="/wiki/Conservation_law" title="Conservation law">conserved quantities</a> can easily be read off from it. The generalized momentum "canonically conjugate to" the coordinate qi is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/732cdc7f806d3627bd7b414c709a7d1d5e9f20f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:10.135ex; height:6.009ex;" alt="{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}.}"> If the Lagrangian L does not depend on some coordinate qi, it follows immediately from the Euler–Lagrange equations that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {p}}_{i}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial L}{\partial q_{i}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {p}}_{i}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial L}{\partial q_{i}}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf88344c6e8d3562636d86fc5e2a422c05675a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:23.997ex; height:6.009ex;" alt="{\displaystyle {\dot {p}}_{i}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial L}{\partial q_{i}}}=0}"> and integrating shows the corresponding generalized momentum equals a constant, a conserved quantity. This is a special case of <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a>. Such coordinates are called "cyclic" or "ignorable". For example, a system may have a Lagrangian <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(r,\theta ,{\dot {s}},{\dot {z}},{\dot {r}},{\dot {\theta }},{\dot {\phi }},t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</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> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(r,\theta ,{\dot {s}},{\dot {z}},{\dot {r}},{\dot {\theta }},{\dot {\phi }},t),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2473d8d50849cd337e63cd2ef6b9dfec8cc43141" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.958ex; height:3.343ex;" alt="{\displaystyle L(r,\theta ,{\dot {s}},{\dot {z}},{\dot {r}},{\dot {\theta }},{\dot {\phi }},t),}"> where r and z are lengths along straight lines, s is an arc length along some curve, and θ and φ are angles. Notice z, s, and φ are all absent in the Lagrangian even though their velocities are not. Then the momenta <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{z}={\frac {\partial L}{\partial {\dot {z}}}},\quad p_{s}={\frac {\partial L}{\partial {\dot {s}}}},\quad p_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}},}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <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> <mi>L</mi> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{z}={\frac {\partial L}{\partial {\dot {z}}}},\quad p_{s}={\frac {\partial L}{\partial {\dot {s}}}},\quad p_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03cc161b33b9d901d2aaa6107fb9d870b1fd8c74" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:34.681ex; height:6.509ex;" alt="{\displaystyle p_{z}={\frac {\partial L}{\partial {\dot {z}}}},\quad p_{s}={\frac {\partial L}{\partial {\dot {s}}}},\quad p_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}},}"> are all conserved quantities. The units and nature of each generalized momentum will depend on the corresponding coordinate; in this case pz is a translational momentum in the z direction, ps is also a translational momentum along the curve s is measured, and pφ is an angular momentum in the plane the angle φ is measured in. However complicated the motion of the system is, all the coordinates and velocities will vary in such a way that these momenta are conserved. <div class="mw-heading mw-heading3"><h3 id="Energy">Energy</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=13" title="Edit section: Energy">edit</a>]</div> Given a Lagrangian <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0efcf5b47104c2ddf893d7608183b766a281cea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.23ex; height:2.509ex;" alt="{\displaystyle L,}"> the <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a> of the corresponding mechanical system is, by definition, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H={\biggl (}\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}{\biggr )}-L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>L</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H={\biggl (}\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}{\biggr )}-L.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e79cafccc1ef42b6f6993084ecba0a7fe6f3329" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.29ex; height:6.843ex;" alt="{\displaystyle H={\biggl (}\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}{\biggr )}-L.}"> This quantity will be equivalent to energy if the generalized coordinates are natural coordinates, i.e., they have no explicit time dependence when expressing position vector: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =\mathbf {r} (q_{1},\cdots ,q_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =\mathbf {r} (q_{1},\cdots ,q_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76780cc9d34e850aec2f1c1cbd2f04584d510c39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.637ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} =\mathbf {r} (q_{1},\cdots ,q_{n})}">. From: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {m}{2}}v^{2}={\frac {m}{2}}\sum _{i,j}\left({\frac {\partial {\vec {r}}}{\partial q_{i}}}{\dot {q}}_{i}\right)\cdot \left({\frac {\partial {\vec {r}}}{\partial q_{j}}}{\dot {q}}_{j}\right)={\frac {m}{2}}\sum _{i,j}a_{ij}{\dot {q}}_{i}{\dot {q}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {m}{2}}v^{2}={\frac {m}{2}}\sum _{i,j}\left({\frac {\partial {\vec {r}}}{\partial q_{i}}}{\dot {q}}_{i}\right)\cdot \left({\frac {\partial {\vec {r}}}{\partial q_{j}}}{\dot {q}}_{j}\right)={\frac {m}{2}}\sum _{i,j}a_{ij}{\dot {q}}_{i}{\dot {q}}_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79fade11af12a99836517a1b55709f8a351e3eab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:58.249ex; height:7.009ex;" alt="{\displaystyle T={\frac {m}{2}}v^{2}={\frac {m}{2}}\sum _{i,j}\left({\frac {\partial {\vec {r}}}{\partial q_{i}}}{\dot {q}}_{i}\right)\cdot \left({\frac {\partial {\vec {r}}}{\partial q_{j}}}{\dot {q}}_{j}\right)={\frac {m}{2}}\sum _{i,j}a_{ij}{\dot {q}}_{i}{\dot {q}}_{j}}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}{\dot {q}}_{k}{\frac {\partial L}{\partial {\dot {q}}_{k}}}=\sum _{k=1}^{n}{\dot {q}}_{k}{\frac {\partial T}{\partial {\dot {q}}_{k}}}={\frac {m}{2}}\left(2\sum _{i,j}a_{ij}{\dot {q}}_{i}{\dot {q}}_{j}\right)=2T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}{\dot {q}}_{k}{\frac {\partial L}{\partial {\dot {q}}_{k}}}=\sum _{k=1}^{n}{\dot {q}}_{k}{\frac {\partial T}{\partial {\dot {q}}_{k}}}={\frac {m}{2}}\left(2\sum _{i,j}a_{ij}{\dot {q}}_{i}{\dot {q}}_{j}\right)=2T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b68a889094ce6452d88ce21ceaa0c3ed4618f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:53.157ex; height:7.676ex;" alt="{\displaystyle \sum _{k=1}^{n}{\dot {q}}_{k}{\frac {\partial L}{\partial {\dot {q}}_{k}}}=\sum _{k=1}^{n}{\dot {q}}_{k}{\frac {\partial T}{\partial {\dot {q}}_{k}}}={\frac {m}{2}}\left(2\sum _{i,j}a_{ij}{\dot {q}}_{i}{\dot {q}}_{j}\right)=2T}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=\left(\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)-L=2T-(T-V)=T+V=E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>L</mi> <mo>=</mo> <mn>2</mn> <mi>T</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo>+</mo> <mi>V</mi> <mo>=</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=\left(\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)-L=2T-(T-V)=T+V=E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af03fdcab77647cf38ddcc854e0a71a3cb7fbc1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.564ex; height:7.509ex;" alt="{\displaystyle H=\left(\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)-L=2T-(T-V)=T+V=E}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}={\frac {\partial \mathbf {r} }{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ij}={\frac {\partial \mathbf {r} }{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{j}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f675c90077f9a368b484d50a1d831e4c601a12d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.576ex; height:6.176ex;" alt="{\displaystyle a_{ij}={\frac {\partial \mathbf {r} }{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{j}}}}"> is a symmetric matrix that is defined for the derivation. <div class="mw-heading mw-heading4"><h4 id="Invariance_under_coordinate_transformations">Invariance under coordinate transformations</h4>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=14" title="Edit section: Invariance under coordinate transformations">edit</a>]</div> At every time instant t, the energy is invariant under <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a> coordinate changes q → Q, i.e. (using natural coordinates) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(\mathbf {q} ,{\dot {\mathbf {q} }},t)=E(\mathbf {Q} ,{\dot {\mathbf {Q} }},t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(\mathbf {q} ,{\dot {\mathbf {q} }},t)=E(\mathbf {Q} ,{\dot {\mathbf {Q} }},t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed0ecc0d17bff21be0dea6ebe7a78fff7c435d7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.577ex; height:3.176ex;" alt="{\displaystyle E(\mathbf {q} ,{\dot {\mathbf {q} }},t)=E(\mathbf {Q} ,{\dot {\mathbf {Q} }},t).}"> Besides this result, the proof below shows that, under such change of coordinates, the derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial L/\partial {\dot {q}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial L/\partial {\dot {q}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80807f4ac7edebc56e5201af71133c6dbe797a35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.558ex; height:2.843ex;" alt="{\displaystyle \partial L/\partial {\dot {q}}_{i}}"> change as coefficients of a linear form. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style="">Proof For a coordinate transformation Q = F(q), we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {Q} =F_{*}(\mathbf {q} )d\mathbf {q} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {Q} =F_{*}(\mathbf {q} )d\mathbf {q} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e596a9b49351ef880540c7cd9b6f8266a4effa58" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.374ex; height:2.843ex;" alt="{\displaystyle d\mathbf {Q} =F_{*}(\mathbf {q} )d\mathbf {q} ,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{*}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{*}(\mathbf {q} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25a617af6512ece7bc7a86b48166d7c62b540767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.774ex; height:2.843ex;" alt="{\displaystyle F_{*}(\mathbf {q} )}"> is the <a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">tangent map</a> of the vector space <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\sum _{i=1}^{n}{\dot {q}}_{i}\cdot \left(\left.{\frac {\partial }{\partial q_{i}}}\right|_{\mathbf {q} }\right)\ {\biggl |}\ {\dot {q}}_{i}\in \mathbb {R} \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow> <mo>(</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\sum _{i=1}^{n}{\dot {q}}_{i}\cdot \left(\left.{\frac {\partial }{\partial q_{i}}}\right|_{\mathbf {q} }\right)\ {\biggl |}\ {\dot {q}}_{i}\in \mathbb {R} \right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a52fec8101fc89435d8f2ad7c6b236ac9041f9f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.787ex; height:7.509ex;" alt="{\displaystyle \left\{\sum _{i=1}^{n}{\dot {q}}_{i}\cdot \left(\left.{\frac {\partial }{\partial q_{i}}}\right|_{\mathbf {q} }\right)\ {\biggl |}\ {\dot {q}}_{i}\in \mathbb {R} \right\}}"> to the vector space <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\sum _{i=1}^{n}{\dot {Q}}_{i}\cdot \left(\left.{\frac {\partial }{\partial Q_{i}}}\right|_{F(\mathbf {q} )}\right)\ {\biggl |}\ {\dot {Q}}_{i}\in \mathbb {R} \right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow> <mo>(</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\sum _{i=1}^{n}{\dot {Q}}_{i}\cdot \left(\left.{\frac {\partial }{\partial Q_{i}}}\right|_{F(\mathbf {q} )}\right)\ {\biggl |}\ {\dot {Q}}_{i}\in \mathbb {R} \right\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe7006ba1ed06fbec4abf65f120b4e995ea5c1a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.055ex; height:7.509ex;" alt="{\displaystyle \left\{\sum _{i=1}^{n}{\dot {Q}}_{i}\cdot \left(\left.{\frac {\partial }{\partial Q_{i}}}\right|_{F(\mathbf {q} )}\right)\ {\biggl |}\ {\dot {Q}}_{i}\in \mathbb {R} \right\},}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle F_{*}(\mathbf {q} )=\left(\left.{\frac {\partial F_{i}}{\partial q_{j}}}\right|_{\mathbf {q} }\right)_{i,j=1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle F_{*}(\mathbf {q} )=\left(\left.{\frac {\partial F_{i}}{\partial q_{j}}}\right|_{\mathbf {q} }\right)_{i,j=1}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be4d85c6a1ad30e3310e2123dbf7505f77bd35cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.658ex; height:6.509ex;" alt="{\displaystyle \textstyle F_{*}(\mathbf {q} )=\left(\left.{\frac {\partial F_{i}}{\partial q_{j}}}\right|_{\mathbf {q} }\right)_{i,j=1}^{n}}"> is the Jacobian. In the coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {q}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {q}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa92ee5d3f74b0d1a40f53393bfec80758d3c2d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.177ex; height:2.676ex;" alt="{\displaystyle {\dot {q}}_{i}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e70ffa505f02650ce3b7062a53c8503551cbcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.285ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{i},}"> the previous formula for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e16bd954789713aa589abd37ac0f2a3c15134bd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.224ex; height:2.509ex;" alt="{\displaystyle d\mathbf {Q} }"> has the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {Q} }}=F_{*}(\mathbf {q} ){\dot {\mathbf {q} }}.}"> <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">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {Q} }}=F_{*}(\mathbf {q} ){\dot {\mathbf {q} }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c6e12a3d91430370b921a66d7e732c203a2d4e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.943ex; height:3.176ex;" alt="{\displaystyle {\dot {\mathbf {Q} }}=F_{*}(\mathbf {q} ){\dot {\mathbf {q} }}.}"> After differentiation involving the product rule, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d{\dot {\mathbf {Q} }}=G(\mathbf {q} ,{\dot {\mathbf {q} }})d\mathbf {q} +F_{*}(\mathbf {q} )d{\dot {\mathbf {q} }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d{\dot {\mathbf {Q} }}=G(\mathbf {q} ,{\dot {\mathbf {q} }})d\mathbf {q} +F_{*}(\mathbf {q} )d{\dot {\mathbf {q} }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f68ad2cabbd7e3cbc7ece75251f3781740f930" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.347ex; height:3.176ex;" alt="{\displaystyle d{\dot {\mathbf {Q} }}=G(\mathbf {q} ,{\dot {\mathbf {q} }})d\mathbf {q} +F_{*}(\mathbf {q} )d{\dot {\mathbf {q} }},}"> where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}G(\mathbf {q} ,{\dot {\mathbf {q} }})d\mathbf {q} &\,{\stackrel {\text{def}}{=}}\,d(F_{*}(\mathbf {q} )){\dot {\mathbf {q} }}=\left(\sum _{k=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }dq_{k}\right)_{i,j=1}^{n}{\dot {\mathbf {q} }}=\left(\sum _{j=1}^{n}{\dot {q}}_{j}\sum _{k=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }dq_{k}\right)_{i=1,\ldots ,n}^{T}\\&=\left(\sum _{k=1}^{n}dq_{k}\sum _{j=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }{\dot {q}}_{j}\right)_{i=1,\ldots ,n}^{T}=\left(\sum _{j=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }{\dot {q}}_{j}\right)_{i,k=1}^{n}d\mathbf {q} .\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> <mi>G</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mtd> <mtd> <mi></mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> </msub> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> </msub> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G(\mathbf {q} ,{\dot {\mathbf {q} }})d\mathbf {q} &\,{\stackrel {\text{def}}{=}}\,d(F_{*}(\mathbf {q} )){\dot {\mathbf {q} }}=\left(\sum _{k=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }dq_{k}\right)_{i,j=1}^{n}{\dot {\mathbf {q} }}=\left(\sum _{j=1}^{n}{\dot {q}}_{j}\sum _{k=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }dq_{k}\right)_{i=1,\ldots ,n}^{T}\\&=\left(\sum _{k=1}^{n}dq_{k}\sum _{j=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }{\dot {q}}_{j}\right)_{i=1,\ldots ,n}^{T}=\left(\sum _{j=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }{\dot {q}}_{j}\right)_{i,k=1}^{n}d\mathbf {q} .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7268ac95e636c71bff930f13d2fcab06370d567" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:89.19ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}G(\mathbf {q} ,{\dot {\mathbf {q} }})d\mathbf {q} &\,{\stackrel {\text{def}}{=}}\,d(F_{*}(\mathbf {q} )){\dot {\mathbf {q} }}=\left(\sum _{k=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }dq_{k}\right)_{i,j=1}^{n}{\dot {\mathbf {q} }}=\left(\sum _{j=1}^{n}{\dot {q}}_{j}\sum _{k=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }dq_{k}\right)_{i=1,\ldots ,n}^{T}\\&=\left(\sum _{k=1}^{n}dq_{k}\sum _{j=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }{\dot {q}}_{j}\right)_{i=1,\ldots ,n}^{T}=\left(\sum _{j=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }{\dot {q}}_{j}\right)_{i,k=1}^{n}d\mathbf {q} .\end{aligned}}}"> In vector notation, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}dL(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)&={\frac {\partial L}{\partial \mathbf {Q} }}d\mathbf {Q} +{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}d{\dot {\mathbf {Q} }}+{\frac {\partial L}{\partial t}}dt\\&=\left({\frac {\partial L}{\partial \mathbf {Q} }}F_{*}(\mathbf {q} )+{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}G(\mathbf {q} ,{\dot {\mathbf {q} }})\right)d\mathbf {q} +{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}F_{*}(\mathbf {q} )d{\dot {\mathbf {q} }}+{\frac {\partial L}{\partial t}}.\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> <mi>d</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> </mrow> </mfrac> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> </mrow> </mfrac> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}dL(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)&={\frac {\partial L}{\partial \mathbf {Q} }}d\mathbf {Q} +{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}d{\dot {\mathbf {Q} }}+{\frac {\partial L}{\partial t}}dt\\&=\left({\frac {\partial L}{\partial \mathbf {Q} }}F_{*}(\mathbf {q} )+{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}G(\mathbf {q} ,{\dot {\mathbf {q} }})\right)d\mathbf {q} +{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}F_{*}(\mathbf {q} )d{\dot {\mathbf {q} }}+{\frac {\partial L}{\partial t}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312657cde5501e8cb067e47d2a0626e43d9bd2f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:69.153ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}dL(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)&={\frac {\partial L}{\partial \mathbf {Q} }}d\mathbf {Q} +{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}d{\dot {\mathbf {Q} }}+{\frac {\partial L}{\partial t}}dt\\&=\left({\frac {\partial L}{\partial \mathbf {Q} }}F_{*}(\mathbf {q} )+{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}G(\mathbf {q} ,{\dot {\mathbf {q} }})\right)d\mathbf {q} +{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}F_{*}(\mathbf {q} )d{\dot {\mathbf {q} }}+{\frac {\partial L}{\partial t}}.\end{aligned}}}"> On the other hand, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dL(\mathbf {q} ,{\dot {\mathbf {q} }},t)={\frac {\partial L}{\partial \mathbf {q} }}d\mathbf {q} +{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}d{\dot {\mathbf {q} }}+{\frac {\partial L}{\partial t}}dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> </mfrac> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dL(\mathbf {q} ,{\dot {\mathbf {q} }},t)={\frac {\partial L}{\partial \mathbf {q} }}d\mathbf {q} +{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}d{\dot {\mathbf {q} }}+{\frac {\partial L}{\partial t}}dt.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb3d3399548fa463d6d66476cf72bc3a768097a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.302ex; height:5.843ex;" alt="{\displaystyle dL(\mathbf {q} ,{\dot {\mathbf {q} }},t)={\frac {\partial L}{\partial \mathbf {q} }}d\mathbf {q} +{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}d{\dot {\mathbf {q} }}+{\frac {\partial L}{\partial t}}dt.}"> It was mentioned earlier that Lagrangians do not depend on the choice of configuration space coordinates, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=L(\mathbf {q} ,{\dot {\mathbf {q} }},t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=L(\mathbf {q} ,{\dot {\mathbf {q} }},t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3552d5ddf024c933a098ea26c9dd0e6cc50579be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.192ex; height:3.176ex;" alt="{\displaystyle L(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=L(\mathbf {q} ,{\dot {\mathbf {q} }},t).}"> One implication of this is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dL(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=dL(\mathbf {q} ,{\dot {\mathbf {q} }},t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dL(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=dL(\mathbf {q} ,{\dot {\mathbf {q} }},t),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4005430933d4878e50bc9ff2f3ff375f97540e22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.623ex; height:3.176ex;" alt="{\displaystyle dL(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=dL(\mathbf {q} ,{\dot {\mathbf {q} }},t),}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}F_{*}(\mathbf {q} )={\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}F_{*}(\mathbf {q} )={\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77d756798c9fa82d3df0e8f465695bd234315c4c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.418ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}F_{*}(\mathbf {q} )={\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}.}"> This demonstrates that, for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {q} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f5466a52a2cbe2cb31294139a2569d83eaa08d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.062ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} ,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {q} }},}"> <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">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {q} }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f69fe5bbcbaeb7e81a2b28491fc9e2f8f3ad203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.062ex; height:2.509ex;" alt="{\displaystyle {\dot {\mathbf {q} }},}"> and <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea3ad87830a1055c7b85c04cf940cfd3b847ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.486ex; height:2.343ex;" alt="{\displaystyle t,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum \limits _{i=1}^{n}{\frac {\partial L}{\partial {\dot {q}}_{i}}}d{\dot {q}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum \limits _{i=1}^{n}{\frac {\partial L}{\partial {\dot {q}}_{i}}}d{\dot {q}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8bb88ac6f9d9aef30382b4910d69d423364075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.871ex; height:6.009ex;" alt="{\displaystyle \textstyle \sum \limits _{i=1}^{n}{\frac {\partial L}{\partial {\dot {q}}_{i}}}d{\dot {q}}_{i}}"> is a well-defined linear form whose coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\partial L}{\partial {\dot {q}}_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\partial L}{\partial {\dot {q}}_{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c53df026a3d820f7219f6e95729f66c338641336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.424ex; height:4.509ex;" alt="{\displaystyle \textstyle {\frac {\partial L}{\partial {\dot {q}}_{i}}}}"> are contravariant 1-tensors. Applying both sides of the equation to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {q} }}}"> <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">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {q} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bedeacdd4e5463759bf946fd6286e8704df1722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.509ex;" alt="{\displaystyle {\dot {\mathbf {q} }}}"> and using the above formula for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {Q} }}}"> <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">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {Q} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f3e8801c9d4d2674efa783255c1ba03c8b90c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:3.009ex;" alt="{\displaystyle {\dot {\mathbf {Q} }}}"> yields <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {Q} }}{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}={\dot {\mathbf {q} }}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}.}"> <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">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {Q} }}{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}={\dot {\mathbf {q} }}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c283947bcace717968a14b9bbc67415547c6e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.068ex; height:6.343ex;" alt="{\displaystyle {\dot {\mathbf {Q} }}{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}={\dot {\mathbf {q} }}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}.}"> The invariance of the energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> follows. </div> <div class="mw-heading mw-heading4"><h4 id="Conservation">Conservation</h4>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=15" title="Edit section: Conservation">edit</a>]</div> In Lagrangian mechanics, the system is <a href="/wiki/Closed_system" title="Closed system">closed</a> if and only if its Lagrangian <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}"> does not explicitly depend on time. The <a href="/wiki/Energy_conservation_law" class="mw-redirect" title="Energy conservation law">energy conservation law</a> states that the energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> of a closed system is an <a href="/wiki/Integral_of_motion" class="mw-redirect" title="Integral of motion">integral of motion</a>. More precisely, let q = q(t) be an extremal. (In other words, q satisfies the Euler–Lagrange equations). Taking the total time-derivative of L along this extremal and using the EL equations leads to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {dL}{dt}}&={\dot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {q} }}+{\ddot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {\dot {q}} }}+{\frac {\partial L}{\partial t}}\\-{\frac {\partial L}{\partial t}}&={\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right){\dot {\mathbf {q} }}+{\ddot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {\dot {q}} }}-{\dot {L}}\\-{\frac {\partial L}{\partial t}}&={\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\mathbf {\dot {q}} -L\right)={\frac {dH}{dt}}\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> <mi>L</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">q</mi> <mo mathvariant="bold">˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <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> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">q</mi> <mo mathvariant="bold">˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">q</mi> <mo mathvariant="bold">˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">q</mi> <mo mathvariant="bold">˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">q</mi> <mo mathvariant="bold">˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>L</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>H</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {dL}{dt}}&={\dot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {q} }}+{\ddot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {\dot {q}} }}+{\frac {\partial L}{\partial t}}\\-{\frac {\partial L}{\partial t}}&={\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right){\dot {\mathbf {q} }}+{\ddot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {\dot {q}} }}-{\dot {L}}\\-{\frac {\partial L}{\partial t}}&={\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\mathbf {\dot {q}} -L\right)={\frac {dH}{dt}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed180e9485aa9fbf4dcf8f94036ba9077e50031e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:34.051ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {dL}{dt}}&={\dot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {q} }}+{\ddot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {\dot {q}} }}+{\frac {\partial L}{\partial t}}\\-{\frac {\partial L}{\partial t}}&={\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right){\dot {\mathbf {q} }}+{\ddot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {\dot {q}} }}-{\dot {L}}\\-{\frac {\partial L}{\partial t}}&={\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\mathbf {\dot {q}} -L\right)={\frac {dH}{dt}}\end{aligned}}}"> If the Lagrangian L does not explicitly depend on time, then ∂L/∂t = 0, then H does not vary with time evolution of particle, indeed, an integral of motion, meaning that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)={\text{constant of time}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant of time</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)={\text{constant of time}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/069e9836a6a4454b531f0ec497fca27a07b9bda4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.878ex; height:2.843ex;" alt="{\displaystyle H(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)={\text{constant of time}}.}"> Hence, if the chosen coordinates were natural coordinates, the energy is conserved. <div class="mw-heading mw-heading4"><h4 id="Kinetic_and_potential_energies">Kinetic and potential energies</h4>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=16" title="Edit section: Kinetic and potential energies">edit</a>]</div> Under all these circumstances,<a href="#cite_note-45">[42]</a> the constant <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=T+V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>T</mi> <mo>+</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=T+V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f4206d30814f7457f644748fa37068ca219be4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.138ex; height:2.343ex;" alt="{\displaystyle E=T+V}"> is the total energy of the system. The kinetic and potential energies still change as the system evolves, but the motion of the system will be such that their sum, the total energy, is constant. This is a valuable simplification, since the energy E is a constant of integration that counts as an arbitrary constant for the problem, and it may be possible to integrate the velocities from this energy relation to solve for the coordinates. <div class="mw-heading mw-heading3"><h3 id="Mechanical_similarity">Mechanical similarity</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=17" title="Edit section: Mechanical similarity">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">Main article: <a href="/wiki/Mechanical_similarity" title="Mechanical similarity">Mechanical similarity</a></div> If the potential energy is a <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous function</a> of the coordinates and independent of time,<a href="#cite_note-46">[43]</a> and all position vectors are scaled by the same nonzero constant α, rk′ = αrk, so that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\alpha \mathbf {r} _{1},\alpha \mathbf {r} _{2},\ldots ,\alpha \mathbf {r} _{N})=\alpha ^{N}V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>α</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>α</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>α</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(\alpha \mathbf {r} _{1},\alpha \mathbf {r} _{2},\ldots ,\alpha \mathbf {r} _{N})=\alpha ^{N}V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f3b296728b8ed6a29e0503e4c6d344d47cc55e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.57ex; height:3.176ex;" alt="{\displaystyle V(\alpha \mathbf {r} _{1},\alpha \mathbf {r} _{2},\ldots ,\alpha \mathbf {r} _{N})=\alpha ^{N}V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N})}"> and time is scaled by a factor β, t′ = βt, then the velocities vk are scaled by a factor of α/β and the kinetic energy T by (α/β)2. The entire Lagrangian has been scaled by the same factor if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha ^{2}}{\beta ^{2}}}=\alpha ^{N}\quad \Rightarrow \quad \beta =\alpha ^{1-{\frac {N}{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <mi>β</mi> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha ^{2}}{\beta ^{2}}}=\alpha ^{N}\quad \Rightarrow \quad \beta =\alpha ^{1-{\frac {N}{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574c4405615079b0ae76bb8b9e4dffbdbf04c2de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.833ex; height:6.343ex;" alt="{\displaystyle {\frac {\alpha ^{2}}{\beta ^{2}}}=\alpha ^{N}\quad \Rightarrow \quad \beta =\alpha ^{1-{\frac {N}{2}}}.}"> Since the lengths and times have been scaled, the trajectories of the particles in the system follow geometrically similar paths differing in size. The length l traversed in time t in the original trajectory corresponds to a new length l′ traversed in time t′ in the new trajectory, given by the ratios <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {t'}{t}}=\left({\frac {l'}{l}}\right)^{1-{\frac {N}{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mo>′</mo> </msup> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>l</mi> <mo>′</mo> </msup> <mi>l</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {t'}{t}}=\left({\frac {l'}{l}}\right)^{1-{\frac {N}{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c65c66f60000df14a6a2dbb725c594a93eb5fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.095ex; height:7.343ex;" alt="{\displaystyle {\frac {t'}{t}}=\left({\frac {l'}{l}}\right)^{1-{\frac {N}{2}}}.}"> <div class="mw-heading mw-heading3"><h3 id="Interacting_particles">Interacting particles</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=18" title="Edit section: Interacting particles">edit</a>]</div> For a given system, if two subsystems A and B are non-interacting, the Lagrangian L of the overall system is the sum of the Lagrangians LA and LB for the subsystems:<a href="#cite_note-Landau_1976_page=4-41">[38]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=L_{A}+L_{B}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=L_{A}+L_{B}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce34e60f35e6d2c435b82b2be229664691ce72a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.279ex; height:2.509ex;" alt="{\displaystyle L=L_{A}+L_{B}.}"> If they do interact this is not possible. In some situations, it may be possible to separate the Lagrangian of the system L into the sum of non-interacting Lagrangians, plus another Lagrangian LAB containing information about the interaction, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=L_{A}+L_{B}+L_{AB}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=L_{A}+L_{B}+L_{AB}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70c1c82bef7edca6d0cee751db3672328857b18" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.414ex; height:2.509ex;" alt="{\displaystyle L=L_{A}+L_{B}+L_{AB}.}"> This may be physically motivated by taking the non-interacting Lagrangians to be kinetic energies only, while the interaction Lagrangian is the system's total potential energy. Also, in the limiting case of negligible interaction, LAB tends to zero reducing to the non-interacting case above. The extension to more than two non-interacting subsystems is straightforward – the overall Lagrangian is the sum of the separate Lagrangians for each subsystem. If there are interactions, then interaction Lagrangians may be added. <div class="mw-heading mw-heading3"><h3 id="Consequences_of_singular_Lagrangians">Consequences of singular Lagrangians</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=19" title="Edit section: Consequences of singular Lagrangians">edit</a>]</div> From the Euler-Lagrange equations, it follows that: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-{\frac {\partial L}{\partial q_{i}}}=0\\&{\frac {\partial ^{2}L}{\partial q_{j}\partial {\dot {q}}_{i}}}{\frac {dq_{j}}{dt}}+{\frac {\partial ^{2}L}{\partial {\dot {q}}_{j}\partial {\dot {q}}_{i}}}{\frac {d{\dot {q}}_{j}}{dt}}+{\frac {\partial L}{\partial t}}-{\frac {\partial L}{\partial q_{i}}}=0\\&\sum _{j}W_{ij}(q,{\dot {q}},t){\ddot {q}}_{j}={\frac {\partial L}{\partial q_{i}}}-{\frac {\partial L}{\partial t}}-\sum _{j}{\frac {\partial ^{2}L}{\partial {\dot {q}}_{i}\partial q_{j}}}{\dot {q}}_{j},\\\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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</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> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-{\frac {\partial L}{\partial q_{i}}}=0\\&{\frac {\partial ^{2}L}{\partial q_{j}\partial {\dot {q}}_{i}}}{\frac {dq_{j}}{dt}}+{\frac {\partial ^{2}L}{\partial {\dot {q}}_{j}\partial {\dot {q}}_{i}}}{\frac {d{\dot {q}}_{j}}{dt}}+{\frac {\partial L}{\partial t}}-{\frac {\partial L}{\partial q_{i}}}=0\\&\sum _{j}W_{ij}(q,{\dot {q}},t){\ddot {q}}_{j}={\frac {\partial L}{\partial q_{i}}}-{\frac {\partial L}{\partial t}}-\sum _{j}{\frac {\partial ^{2}L}{\partial {\dot {q}}_{i}\partial q_{j}}}{\dot {q}}_{j},\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed96cdcf52a04c5bfb0fc145d25fa9f24fb66432" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:48.781ex; height:20.176ex;" alt="{\displaystyle {\begin{aligned}&{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-{\frac {\partial L}{\partial q_{i}}}=0\\&{\frac {\partial ^{2}L}{\partial q_{j}\partial {\dot {q}}_{i}}}{\frac {dq_{j}}{dt}}+{\frac {\partial ^{2}L}{\partial {\dot {q}}_{j}\partial {\dot {q}}_{i}}}{\frac {d{\dot {q}}_{j}}{dt}}+{\frac {\partial L}{\partial t}}-{\frac {\partial L}{\partial q_{i}}}=0\\&\sum _{j}W_{ij}(q,{\dot {q}},t){\ddot {q}}_{j}={\frac {\partial L}{\partial q_{i}}}-{\frac {\partial L}{\partial t}}-\sum _{j}{\frac {\partial ^{2}L}{\partial {\dot {q}}_{i}\partial q_{j}}}{\dot {q}}_{j},\\\end{aligned}}}"> where the matrix is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{ij}={\frac {\partial ^{2}L}{\partial {\dot {q}}_{i}\partial {\dot {q}}_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{ij}={\frac {\partial ^{2}L}{\partial {\dot {q}}_{i}\partial {\dot {q}}_{j}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d5900f349bf99931fbdedcf2661c377a95f75f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.705ex; height:6.676ex;" alt="{\displaystyle W_{ij}={\frac {\partial ^{2}L}{\partial {\dot {q}}_{i}\partial {\dot {q}}_{j}}}}">. If the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"> is non-singular, the above equations can be solved to represent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb87f34d76df7d2c559c52a8e193900cb4d5558" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.377ex; height:2.509ex;" alt="{\displaystyle {\ddot {q}}}"> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\dot {q}},q,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\dot {q}},q,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70421640fa7083c61fc84b74e3b216a95745e519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.163ex; height:2.843ex;" alt="{\displaystyle ({\dot {q}},q,t)}">. If the matrix is non-invertible, it would not be possible to represent all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb87f34d76df7d2c559c52a8e193900cb4d5558" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.377ex; height:2.509ex;" alt="{\displaystyle {\ddot {q}}}">'s as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\dot {q}},q,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\dot {q}},q,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70421640fa7083c61fc84b74e3b216a95745e519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.163ex; height:2.843ex;" alt="{\displaystyle ({\dot {q}},q,t)}"> but also, the Hamiltonian equations of motions will not take the standard form.<a href="#cite_note-47">[44]</a> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=20" title="Edit section: Examples">edit</a>]</div> The following examples apply Lagrange's equations of the second kind to mechanical problems. <div class="mw-heading mw-heading3"><h3 id="Conservative_force">Conservative force</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=21" title="Edit section: Conservative force">edit</a>]</div> A particle of mass m moves under the influence of a <a href="/wiki/Conservative_force" title="Conservative force">conservative force</a> derived from the <a href="/wiki/Gradient" title="Gradient">gradient</a> ∇ of a <a href="/wiki/Scalar_potential" title="Scalar potential">scalar potential</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =-{\boldsymbol {\nabla }}V(\mathbf {r} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =-{\boldsymbol {\nabla }}V(\mathbf {r} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e136fd887cfe99ae7dc3d5afccd543c342c65d1c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.161ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} =-{\boldsymbol {\nabla }}V(\mathbf {r} ).}"> If there are more particles, in accordance with the above results, the total kinetic energy is a sum over all the particle kinetic energies, and the potential is a function of all the coordinates. <div class="mw-heading mw-heading4"><h4 id="Cartesian_coordinates">Cartesian coordinates</h4>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=22" title="Edit section: Cartesian coordinates">edit</a>]</div> The Lagrangian of the particle can be written <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x,y,z,{\dot {x}},{\dot {y}},{\dot {z}})={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})-V(x,y,z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(x,y,z,{\dot {x}},{\dot {y}},{\dot {z}})={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})-V(x,y,z).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8c3e24b80716a5d3b8426200e9b701ef3b2e3b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.507ex; height:5.176ex;" alt="{\displaystyle L(x,y,z,{\dot {x}},{\dot {y}},{\dot {z}})={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})-V(x,y,z).}"> The equations of motion for the particle are found by applying the <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equation</a>, for the x coordinate <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)={\frac {\partial L}{\partial x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)={\frac {\partial L}{\partial x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a71de9c538bdff5a2c623a500f3e20ee8744f5c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.996ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)={\frac {\partial L}{\partial x}},}"> with derivatives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial L}{\partial x}}=-{\frac {\partial V}{\partial x}},\quad {\frac {\partial L}{\partial {\dot {x}}}}=m{\dot {x}},\quad {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)=m{\ddot {x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</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> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial L}{\partial x}}=-{\frac {\partial V}{\partial x}},\quad {\frac {\partial L}{\partial {\dot {x}}}}=m{\dot {x}},\quad {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)=m{\ddot {x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066c1275d55718a3eb24bdfee45700b24632360" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.133ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial L}{\partial x}}=-{\frac {\partial V}{\partial x}},\quad {\frac {\partial L}{\partial {\dot {x}}}}=m{\dot {x}},\quad {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)=m{\ddot {x}},}"> hence <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\ddot {x}}=-{\frac {\partial V}{\partial x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </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> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\ddot {x}}=-{\frac {\partial V}{\partial x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f29905aaf164cbf0c70f8757713ddb2a463ecbe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.865ex; height:5.509ex;" alt="{\displaystyle m{\ddot {x}}=-{\frac {\partial V}{\partial x}},}"> and similarly for the y and z coordinates. Collecting the equations in vector form we find <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\ddot {\mathbf {r} }}=-{\boldsymbol {\nabla }}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\ddot {\mathbf {r} }}=-{\boldsymbol {\nabla }}V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6887fd7b6aa83fe41b2458e47b70b1758111125" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.123ex; height:2.343ex;" alt="{\displaystyle m{\ddot {\mathbf {r} }}=-{\boldsymbol {\nabla }}V}"> which is <a href="/wiki/Newton%27s_second_law_of_motion" class="mw-redirect" title="Newton's second law of motion">Newton's second law of motion</a> for a particle subject to a conservative force. <div class="mw-heading mw-heading4"><h4 id="Polar_coordinates_in_2D_and_3D">Polar coordinates in 2D and 3D</h4>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=23" title="Edit section: Polar coordinates in 2D and 3D">edit</a>]</div> Using the spherical coordinates (r, θ, φ) as commonly used in physics (ISO 80000-2:2019 convention), where r is the radial distance to origin, θ is polar angle (also known as colatitude, zenith angle, normal angle, or inclination angle), and φ is the azimuthal angle, the Lagrangian for a central potential is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\frac {m}{2}}({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+r^{2}\sin ^{2}\theta \,{\dot {\varphi }}^{2})-V(r).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <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>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <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 stretchy="false">)</mo> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {m}{2}}({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+r^{2}\sin ^{2}\theta \,{\dot {\varphi }}^{2})-V(r).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a22ac51d811775df1ffdf7db79e2359950bbc4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.893ex; height:4.676ex;" alt="{\displaystyle L={\frac {m}{2}}({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+r^{2}\sin ^{2}\theta \,{\dot {\varphi }}^{2})-V(r).}"> So, in spherical coordinates, the Euler–Lagrange equations are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\ddot {r}}-mr({\dot {\theta }}^{2}+\sin ^{2}\theta \,{\dot {\varphi }}^{2})+{\frac {\partial V}{\partial r}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>m</mi> <mi>r</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <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 stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\ddot {r}}-mr({\dot {\theta }}^{2}+\sin ^{2}\theta \,{\dot {\varphi }}^{2})+{\frac {\partial V}{\partial r}}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4fab2d65969d3975d2984ec334f16440d66f35" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.374ex; height:5.509ex;" alt="{\displaystyle m{\ddot {r}}-mr({\dot {\theta }}^{2}+\sin ^{2}\theta \,{\dot {\varphi }}^{2})+{\frac {\partial V}{\partial r}}=0,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}{\dot {\theta }})-mr^{2}\sin \theta \cos \theta \,{\dot {\varphi }}^{2}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</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>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <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> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}{\dot {\theta }})-mr^{2}\sin \theta \cos \theta \,{\dot {\varphi }}^{2}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d03cd05305aab021433002e72f9daf01e0b819" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.84ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}{\dot {\theta }})-mr^{2}\sin \theta \cos \theta \,{\dot {\varphi }}^{2}=0,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}\sin ^{2}\theta \,{\dot {\varphi }})=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}\sin ^{2}\theta \,{\dot {\varphi }})=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/623b16f9cf6185db17ab7439093c6a41f9b7ee2a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.525ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}\sin ^{2}\theta \,{\dot {\varphi }})=0.}"> The φ coordinate is cyclic since it does not appear in the Lagrangian, so the conserved momentum in the system is the angular momentum <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\varphi }={\frac {\partial L}{\partial {\dot {\varphi }}}}=mr^{2}\sin ^{2}\theta {\dot {\varphi }},}"> <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> <mi>L</mi> </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> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\varphi }={\frac {\partial L}{\partial {\dot {\varphi }}}}=mr^{2}\sin ^{2}\theta {\dot {\varphi }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92b9cd7b495d48834f1bf3e8455f8d28627c9d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:24.599ex; height:6.009ex;" alt="{\displaystyle p_{\varphi }={\frac {\partial L}{\partial {\dot {\varphi }}}}=mr^{2}\sin ^{2}\theta {\dot {\varphi }},}"> in which r, θ and dφ/dt can all vary with time, but only in such a way that pφ is constant. The Lagrangian in two-dimensional polar coordinates is recovered by fixing θ to the constant value π/2. <div class="mw-heading mw-heading3"><h3 id="Pendulum_on_a_movable_support">Pendulum on a movable support</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=24" title="Edit section: Pendulum on a movable support">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:PendulumWithMovableSupport.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/PendulumWithMovableSupport.svg/250px-PendulumWithMovableSupport.svg.png" decoding="async" width="250" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/PendulumWithMovableSupport.svg/375px-PendulumWithMovableSupport.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/PendulumWithMovableSupport.svg/500px-PendulumWithMovableSupport.svg.png 2x" data-file-width="195" data-file-height="178" /></a><figcaption>Sketch of the situation with definition of the coordinates (click to enlarge)</figcaption></figure> Consider a pendulum of mass m and length ℓ, which is attached to a support with mass M, which can move along a line in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-direction. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle <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 }"> from the vertical. The coordinates and velocity components of the pendulum bob are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rll}&x_{\mathrm {pend} }=x+\ell \sin \theta &\quad \Rightarrow \quad {\dot {x}}_{\mathrm {pend} }={\dot {x}}+\ell {\dot {\theta }}\cos \theta \\&y_{\mathrm {pend} }=-\ell \cos \theta &\quad \Rightarrow \quad {\dot {y}}_{\mathrm {pend} }=\ell {\dot {\theta }}\sin \theta .\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>ℓ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>ℓ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rll}&x_{\mathrm {pend} }=x+\ell \sin \theta &\quad \Rightarrow \quad {\dot {x}}_{\mathrm {pend} }={\dot {x}}+\ell {\dot {\theta }}\cos \theta \\&y_{\mathrm {pend} }=-\ell \cos \theta &\quad \Rightarrow \quad {\dot {y}}_{\mathrm {pend} }=\ell {\dot {\theta }}\sin \theta .\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f96bff196aef9c2de777db51ef73acd5d601a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.607ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{rll}&x_{\mathrm {pend} }=x+\ell \sin \theta &\quad \Rightarrow \quad {\dot {x}}_{\mathrm {pend} }={\dot {x}}+\ell {\dot {\theta }}\cos \theta \\&y_{\mathrm {pend} }=-\ell \cos \theta &\quad \Rightarrow \quad {\dot {y}}_{\mathrm {pend} }=\ell {\dot {\theta }}\sin \theta .\end{array}}}"> The generalized coordinates can be taken to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> 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 }">. The kinetic energy of the system is then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left({\dot {x}}_{\mathrm {pend} }^{2}+{\dot {y}}_{\mathrm {pend} }^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>M</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left({\dot {x}}_{\mathrm {pend} }^{2}+{\dot {y}}_{\mathrm {pend} }^{2}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85d06640b01fe80841a1a786446a4b7c3f8def30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.836ex; height:5.176ex;" alt="{\displaystyle T={\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left({\dot {x}}_{\mathrm {pend} }^{2}+{\dot {y}}_{\mathrm {pend} }^{2}\right)}"> and the potential energy is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=mgy_{\mathrm {pend} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>m</mi> <mi>g</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=mgy_{\mathrm {pend} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ec4ff69448104cc7d4260bd3ff8cdb2dc956880" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.885ex; height:2.843ex;" alt="{\displaystyle V=mgy_{\mathrm {pend} }}"> giving the Lagrangian <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcl}L&=&T-V\\&=&{\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left[\left({\dot {x}}+\ell {\dot {\theta }}\cos \theta \right)^{2}+\left(\ell {\dot {\theta }}\sin \theta \right)^{2}\right]+mg\ell \cos \theta \\&=&{\frac {1}{2}}\left(M+m\right){\dot {x}}^{2}+m{\dot {x}}\ell {\dot {\theta }}\cos \theta +{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}+mg\ell \cos \theta .\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>L</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>T</mi> <mo>−</mo> <mi>V</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>M</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>m</mi> <mi>g</mi> <mi>ℓ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>M</mi> <mo>+</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>m</mi> <mi>g</mi> <mi>ℓ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}L&=&T-V\\&=&{\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left[\left({\dot {x}}+\ell {\dot {\theta }}\cos \theta \right)^{2}+\left(\ell {\dot {\theta }}\sin \theta \right)^{2}\right]+mg\ell \cos \theta \\&=&{\frac {1}{2}}\left(M+m\right){\dot {x}}^{2}+m{\dot {x}}\ell {\dot {\theta }}\cos \theta +{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}+mg\ell \cos \theta .\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/495fcfe1fd6e2433a1b2166884ff19686152137f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:65.612ex; height:13.843ex;" alt="{\displaystyle {\begin{array}{rcl}L&=&T-V\\&=&{\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left[\left({\dot {x}}+\ell {\dot {\theta }}\cos \theta \right)^{2}+\left(\ell {\dot {\theta }}\sin \theta \right)^{2}\right]+mg\ell \cos \theta \\&=&{\frac {1}{2}}\left(M+m\right){\dot {x}}^{2}+m{\dot {x}}\ell {\dot {\theta }}\cos \theta +{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}+mg\ell \cos \theta .\end{array}}}"> Since x is absent from the Lagrangian, it is a cyclic coordinate. The conserved momentum is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{x}={\frac {\partial L}{\partial {\dot {x}}}}=(M+m){\dot {x}}+m\ell {\dot {\theta }}\cos \theta ,}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>m</mi> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{x}={\frac {\partial L}{\partial {\dot {x}}}}=(M+m){\dot {x}}+m\ell {\dot {\theta }}\cos \theta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ede4811533c38fd5b297061f41824f8bcb397ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:35.656ex; height:5.509ex;" alt="{\displaystyle p_{x}={\frac {\partial L}{\partial {\dot {x}}}}=(M+m){\dot {x}}+m\ell {\dot {\theta }}\cos \theta ,}"> and the Lagrange equation for the support coordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M+m){\ddot {x}}+m\ell {\ddot {\theta }}\cos \theta -m\ell {\dot {\theta }}^{2}\sin \theta =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>m</mi> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>m</mi> <mi>ℓ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M+m){\ddot {x}}+m\ell {\ddot {\theta }}\cos \theta -m\ell {\dot {\theta }}^{2}\sin \theta =0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d68d58becf9b09d7d923281f0a9f54458133f651" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.533ex; height:3.676ex;" alt="{\displaystyle (M+m){\ddot {x}}+m\ell {\ddot {\theta }}\cos \theta -m\ell {\dot {\theta }}^{2}\sin \theta =0.}"> The Lagrange equation for the angle θ is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left[m({\dot {x}}\ell \cos \theta +\ell ^{2}{\dot {\theta }})\right]+m\ell ({\dot {x}}{\dot {\theta }}+g)\sin \theta =0;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>ℓ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>m</mi> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left[m({\dot {x}}\ell \cos \theta +\ell ^{2}{\dot {\theta }})\right]+m\ell ({\dot {x}}{\dot {\theta }}+g)\sin \theta =0;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27ef95240aa335e09f85c8c7ef67a7301547c588" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.825ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left[m({\dot {x}}\ell \cos \theta +\ell ^{2}{\dot {\theta }})\right]+m\ell ({\dot {x}}{\dot {\theta }}+g)\sin \theta =0;}"> and simplifying <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {\theta }}+{\frac {\ddot {x}}{\ell }}\cos \theta +{\frac {g}{\ell }}\sin \theta =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> <mi>ℓ</mi> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mi>ℓ</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {\theta }}+{\frac {\ddot {x}}{\ell }}\cos \theta +{\frac {g}{\ell }}\sin \theta =0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180061615316c3fba1363de4da7d1fb243401613" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.758ex; height:5.343ex;" alt="{\displaystyle {\ddot {\theta }}+{\frac {\ddot {x}}{\ell }}\cos \theta +{\frac {g}{\ell }}\sin \theta =0.}"> These equations may look quite complicated, but finding them with Newton's laws would have required carefully identifying all forces, which would have been much more laborious and prone to errors. By considering limit cases, the correctness of this system can be verified: For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {x}}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {x}}\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14b5ffbe4d5cab84a3fe4c43a5da25dc5d39cd9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.106ex; height:2.176ex;" alt="{\displaystyle {\ddot {x}}\to 0}"> should give the equations of motion for a <a href="/wiki/Simple_pendulum" class="mw-redirect" title="Simple pendulum">simple pendulum</a> that is at rest in some <a href="/wiki/Inertial_frame" class="mw-redirect" title="Inertial frame">inertial frame</a>, while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {\theta }}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {\theta }}\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9dc14b48aa263f561e813ba7dd2add570e3c8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.132ex; height:2.676ex;" alt="{\displaystyle {\ddot {\theta }}\to 0}"> should give the equations for a pendulum in a constantly accelerating system, etc. Furthermore, it is trivial to obtain the results numerically, given suitable starting conditions and a chosen time step, by <a href="/wiki/Numerical_ordinary_differential_equations" class="mw-redirect" title="Numerical ordinary differential equations">stepping through the results iteratively</a>. <div class="mw-heading mw-heading3"><h3 id="Two-body_central_force_problem">Two-body central force problem</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=25" title="Edit section: Two-body central force problem">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/Two-body_problem" title="Two-body problem">Two-body problem</a> and <a href="/wiki/Central_force" title="Central force">Central force</a></div> Two bodies of masses m1 and m2 with position vectors r1 and r2 are in orbit about each other due to an attractive <a href="/wiki/Central_potential" class="mw-redirect" title="Central potential">central potential</a> V. We may write down the Lagrangian in terms of the position coordinates as they are, but it is an established procedure to convert the two-body problem into a one-body problem as follows. Introduce the <a href="/wiki/Jacobi_coordinates" title="Jacobi coordinates">Jacobi coordinates</a>; the separation of the bodies r = r2 − r1 and the location of the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> R = (m1r1 + m2r2)/(m1 + m2). The Lagrangian is then<a href="#cite_note-48">[45]</a><a href="#cite_note-49">[46]</a><a href="#cite_note-50">[nb 4]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\underbrace {{\frac {1}{2}}M{\dot {\mathbf {R} }}^{2}} _{L_{\text{cm}}}+\underbrace {{\frac {1}{2}}\mu {\dot {\mathbf {r} }}^{2}-V(|\mathbf {r} |)} _{L_{\text{rel}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>M</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>cm</mtext> </mrow> </msub> </mrow> </munder> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>μ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\underbrace {{\frac {1}{2}}M{\dot {\mathbf {R} }}^{2}} _{L_{\text{cm}}}+\underbrace {{\frac {1}{2}}\mu {\dot {\mathbf {r} }}^{2}-V(|\mathbf {r} |)} _{L_{\text{rel}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/968e585f8fc125b9c8a442482d0caff0290d0025" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; margin-right: -0.028ex; width:29.497ex; height:9.009ex;" alt="{\displaystyle L=\underbrace {{\frac {1}{2}}M{\dot {\mathbf {R} }}^{2}} _{L_{\text{cm}}}+\underbrace {{\frac {1}{2}}\mu {\dot {\mathbf {r} }}^{2}-V(|\mathbf {r} |)} _{L_{\text{rel}}}}"> where M = m1 + m2 is the total mass, μ = m1m2/(m1 + m2) is the <a href="/wiki/Reduced_mass" title="Reduced mass">reduced mass</a>, and V the potential of the radial force, which depends only on the <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">magnitude</a> of the separation |r| = |r2 − r1|. The Lagrangian splits into a center-of-mass term Lcm and a relative motion term Lrel. The Euler–Lagrange equation for R is simply <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M{\ddot {\mathbf {R} }}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M{\ddot {\mathbf {R} }}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c8cbc4e9437182f1c68b2185037442ddc0c0db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.353ex; height:3.009ex;" alt="{\displaystyle M{\ddot {\mathbf {R} }}=0,}"> which states the center of mass moves in a straight line at constant velocity. Since the relative motion only depends on the magnitude of the separation, it is ideal to use polar coordinates (r, θ) and take r = |r|, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\text{rel}}={\frac {1}{2}}\mu \left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)-V(r),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>μ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <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>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\text{rel}}={\frac {1}{2}}\mu \left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)-V(r),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1655037c1a9550accef5e79f7008c6a73eef252" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.14ex; height:5.176ex;" alt="{\displaystyle L_{\text{rel}}={\frac {1}{2}}\mu \left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)-V(r),}"> so θ is a cyclic coordinate with the corresponding conserved (angular) momentum <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\theta }={\frac {\partial L_{\text{rel}}}{\partial {\dot {\theta }}}}=\mu r^{2}{\dot {\theta }}=\ell .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>μ</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>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>ℓ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\theta }={\frac {\partial L_{\text{rel}}}{\partial {\dot {\theta }}}}=\mu r^{2}{\dot {\theta }}=\ell .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03d4c9245ffb12dea45cd0a7bb9a22020269a1f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:23.836ex; height:6.009ex;" alt="{\displaystyle p_{\theta }={\frac {\partial L_{\text{rel}}}{\partial {\dot {\theta }}}}=\mu r^{2}{\dot {\theta }}=\ell .}"> The radial coordinate r and angular velocity dθ/dt can vary with time, but only in such a way that ℓ is constant. The Lagrange equation for r is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu r{\dot {\theta }}^{2}-{\frac {dV}{dr}}=\mu {\ddot {r}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>V</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu r{\dot {\theta }}^{2}-{\frac {dV}{dr}}=\mu {\ddot {r}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7959cce4d4323543a47213d9d52e54e3e727f71c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.979ex; height:5.509ex;" alt="{\displaystyle \mu r{\dot {\theta }}^{2}-{\frac {dV}{dr}}=\mu {\ddot {r}}.}"> This equation is identical to the radial equation obtained using Newton's laws in a co-rotating reference frame, that is, a frame rotating with the reduced mass so it appears stationary. Eliminating the angular velocity dθ/dt from this radial equation,<a href="#cite_note-51">[47]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu {\ddot {r}}=-{\frac {\mathrm {d} V}{\mathrm {d} r}}+{\frac {\ell ^{2}}{\mu r^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>μ</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu {\ddot {r}}=-{\frac {\mathrm {d} V}{\mathrm {d} r}}+{\frac {\ell ^{2}}{\mu r^{3}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db58016a28abee46a3b0c532ecb7100aaae03824" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.344ex; height:6.343ex;" alt="{\displaystyle \mu {\ddot {r}}=-{\frac {\mathrm {d} V}{\mathrm {d} r}}+{\frac {\ell ^{2}}{\mu r^{3}}}.}"> which is the equation of motion for a one-dimensional problem in which a particle of mass μ is subjected to the inward central force −dV/dr and a second outward force, called in this context the (Lagrangian) centrifugal force (see <a href="/wiki/Centrifugal_force#Other_uses_of_the_term" title="Centrifugal force">centrifugal force#Other uses of the term</a>): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mathrm {cf} }=\mu r{\dot {\theta }}^{2}={\frac {\ell ^{2}}{\mu r^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>μ</mi> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>μ</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mathrm {cf} }=\mu r{\dot {\theta }}^{2}={\frac {\ell ^{2}}{\mu r^{3}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2d3c6d2cfa959b05c92d7a96f927bee436f896" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.114ex; height:6.343ex;" alt="{\displaystyle F_{\mathrm {cf} }=\mu r{\dot {\theta }}^{2}={\frac {\ell ^{2}}{\mu r^{3}}}.}"> Of course, if one remains entirely within the one-dimensional formulation, ℓ enters only as some imposed parameter of the external outward force, and its interpretation as angular momentum depends upon the more general two-dimensional problem from which the one-dimensional problem originated. If one arrives at this equation using Newtonian mechanics in a co-rotating frame, the interpretation is evident as the centrifugal force in that frame due to the rotation of the frame itself. If one arrives at this equation directly by using the generalized coordinates (r, θ) and simply following the Lagrangian formulation without thinking about frames at all, the interpretation is that the centrifugal force is an outgrowth of using polar coordinates. As Hildebrand says:<a href="#cite_note-52">[48]</a> "Since such quantities are not true physical forces, they are often called inertia forces. Their presence or absence depends, not upon the particular problem at hand, but upon the coordinate system chosen." In particular, if Cartesian coordinates are chosen, the centrifugal force disappears, and the formulation involves only the central force itself, which provides the <a href="/wiki/Centripetal_force" title="Centripetal force">centripetal force</a> for a curved motion. This viewpoint, that fictitious forces originate in the choice of coordinates, often is expressed by users of the Lagrangian method. This view arises naturally in the Lagrangian approach, because the frame of reference is (possibly unconsciously) selected by the choice of coordinates. For example, see<a href="#cite_note-53">[49]</a> for a comparison of Lagrangians in an inertial and in a noninertial frame of reference. See also the discussion of "total" and "updated" Lagrangian formulations in.<a href="#cite_note-54">[50]</a> Unfortunately, this usage of "inertial force" conflicts with the Newtonian idea of an inertial force. In the Newtonian view, an inertial force originates in the acceleration of the frame of observation (the fact that it is not an <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial frame of reference</a>), not in the choice of coordinate system. To keep matters clear, it is safest to refer to the Lagrangian inertial forces as generalized inertial forces, to distinguish them from the Newtonian vector inertial forces. That is, one should avoid following Hildebrand when he says (p. 155) "we deal always with generalized forces, velocities accelerations, and momenta. For brevity, the adjective "generalized" will be omitted frequently." It is known that the Lagrangian of a system is not unique. Within the Lagrangian formalism the Newtonian fictitious forces can be identified by the existence of alternative Lagrangians in which the fictitious forces disappear, sometimes found by exploiting the symmetry of the system.<a href="#cite_note-55">[51]</a> <div class="mw-heading mw-heading2"><h2 id="Extensions_to_include_non-conservative_forces">Extensions to include non-conservative forces</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=26" title="Edit section: Extensions to include non-conservative forces">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Dissipative_forces">Dissipative forces</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=27" title="Edit section: Dissipative forces">edit</a>]</div> <a href="/wiki/Dissipation" title="Dissipation">Dissipation</a> (i.e. non-conservative systems) can also be treated with an effective Lagrangian formulated by a certain doubling of the degrees of freedom.<a href="#cite_note-56">[52]</a><a href="#cite_note-57">[53]</a><a href="#cite_note-58">[54]</a><a href="#cite_note-59">[55]</a> In a more general formulation, the forces could be both conservative and <a href="/wiki/Viscosity" title="Viscosity">viscous</a>. If an appropriate transformation can be found from the Fi, <a href="/wiki/John_Strutt,_3rd_Baron_Rayleigh" class="mw-redirect" title="John Strutt, 3rd Baron Rayleigh">Rayleigh</a> suggests using a <a href="/wiki/Rayleigh_dissipation_function" title="Rayleigh dissipation function">dissipation function</a>, D, of the following form:<a href="#cite_note-Torby_1984_page=271-60">[56]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\frac {1}{2}}\sum _{j=1}^{m}\sum _{k=1}^{m}C_{jk}{\dot {q}}_{j}{\dot {q}}_{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D={\frac {1}{2}}\sum _{j=1}^{m}\sum _{k=1}^{m}C_{jk}{\dot {q}}_{j}{\dot {q}}_{k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbd465ad6df19616fe47dbf3a512f1f77b3c8c18" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.72ex; height:7.176ex;" alt="{\displaystyle D={\frac {1}{2}}\sum _{j=1}^{m}\sum _{k=1}^{m}C_{jk}{\dot {q}}_{j}{\dot {q}}_{k},}"> where Cjk are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them. If D is defined this way, then<a href="#cite_note-Torby_1984_page=271-60">[56]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{j}=-{\frac {\partial V}{\partial q_{j}}}-{\frac {\partial D}{\partial {\dot {q}}_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <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> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>D</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{j}=-{\frac {\partial V}{\partial q_{j}}}-{\frac {\partial D}{\partial {\dot {q}}_{j}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f26c164d31b8a830b1703cb770a2c2bfa730cbf5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.037ex; height:6.343ex;" alt="{\displaystyle Q_{j}=-{\frac {\partial V}{\partial q_{j}}}-{\frac {\partial D}{\partial {\dot {q}}_{j}}}}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial L}{\partial q_{j}}}+{\frac {\partial D}{\partial {\dot {q}}_{j}}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>D</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial L}{\partial q_{j}}}+{\frac {\partial D}{\partial {\dot {q}}_{j}}}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc630cb8c0be1650303f59a85083debd5f06cd0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.608ex; height:7.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial L}{\partial q_{j}}}+{\frac {\partial D}{\partial {\dot {q}}_{j}}}=0.}"> <div class="mw-heading mw-heading3"><h3 id="Electromagnetism">Electromagnetism</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=28" title="Edit section: Electromagnetism">edit</a>]</div> A test particle is a particle whose <a href="/wiki/Mass" title="Mass">mass</a> and <a href="/wiki/Electric_charge" title="Electric charge">charge</a> are assumed to be so small that its effect on external system is insignificant. It is often a hypothetical simplified point particle with no properties other than mass and charge. Real particles like <a href="/wiki/Electron" title="Electron">electrons</a> and <a href="/wiki/Up_quark" title="Up quark">up quarks</a> are more complex and have additional terms in their Lagrangians. Not only can the fields form non conservative potentials, these potentials can also be velocity dependent. The Lagrangian for a <a href="/wiki/Charged_particle" title="Charged particle">charged particle</a> with <a href="/wiki/Electrical_charge" class="mw-redirect" title="Electrical charge">electrical charge</a> q, interacting with an <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a>, is the prototypical example of a velocity-dependent potential. The electric <a href="/wiki/Scalar_potential" title="Scalar potential">scalar potential</a> ϕ = ϕ(r, t) and <a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">magnetic vector potential</a> A = A(r, t) are defined from the <a href="/wiki/Electric_field" title="Electric field">electric field</a> E = E(r, t) and <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> B = B(r, t) as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} =-{\boldsymbol {\nabla }}\phi -{\frac {\partial \mathbf {A} }{\partial t}},\quad \mathbf {B} ={\boldsymbol {\nabla }}\times \mathbf {A} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>ϕ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =-{\boldsymbol {\nabla }}\phi -{\frac {\partial \mathbf {A} }{\partial t}},\quad \mathbf {B} ={\boldsymbol {\nabla }}\times \mathbf {A} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c3d6f858a02684dd6dea11c3d50ab146b45b352" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.378ex; height:5.509ex;" alt="{\displaystyle \mathbf {E} =-{\boldsymbol {\nabla }}\phi -{\frac {\partial \mathbf {A} }{\partial t}},\quad \mathbf {B} ={\boldsymbol {\nabla }}\times \mathbf {A} .}"> The Lagrangian of a massive charged test particle in an electromagnetic field <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\tfrac {1}{2}}m{\dot {\mathbf {r} }}^{2}+q\,{\dot {\mathbf {r} }}\cdot \mathbf {A} -q\phi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>q</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>−</mo> <mi>q</mi> <mi>ϕ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\tfrac {1}{2}}m{\dot {\mathbf {r} }}^{2}+q\,{\dot {\mathbf {r} }}\cdot \mathbf {A} -q\phi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e02202487b2e38bdf5a33245cc63a30971061923" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.697ex; height:3.509ex;" alt="{\displaystyle L={\tfrac {1}{2}}m{\dot {\mathbf {r} }}^{2}+q\,{\dot {\mathbf {r} }}\cdot \mathbf {A} -q\phi ,}"> is called <a href="/wiki/Minimal_coupling" title="Minimal coupling">minimal coupling</a>. This is a good example of when the common <a href="/wiki/Rule_of_thumb" title="Rule of thumb">rule of thumb</a> that the Lagrangian is the kinetic energy minus the potential energy is incorrect. Combined with <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equation</a>, it produces the <a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a> law <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\ddot {\mathbf {r} }}=q\mathbf {E} +q{\dot {\mathbf {r} }}\times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\ddot {\mathbf {r} }}=q\mathbf {E} +q{\dot {\mathbf {r} }}\times \mathbf {B} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59545cd02a3116f34154971fb43ed142f93efde9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.942ex; height:2.509ex;" alt="{\displaystyle m{\ddot {\mathbf {r} }}=q\mathbf {E} +q{\dot {\mathbf {r} }}\times \mathbf {B} }"> Under <a href="/wiki/Gauge_transformation" class="mw-redirect" title="Gauge transformation">gauge transformation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \rightarrow \mathbf {A} +{\boldsymbol {\nabla }}f,\quad \phi \rightarrow \phi -{\dot {f}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>f</mi> <mo>,</mo> <mspace width="1em" /> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi>ϕ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \rightarrow \mathbf {A} +{\boldsymbol {\nabla }}f,\quad \phi \rightarrow \phi -{\dot {f}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b508a8c79807013be44a303abb8a2ba54f927978" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.926ex; height:3.176ex;" alt="{\displaystyle \mathbf {A} \rightarrow \mathbf {A} +{\boldsymbol {\nabla }}f,\quad \phi \rightarrow \phi -{\dot {f}},}"> where f(r,t) is any scalar function of space and time, the aforementioned Lagrangian transforms like: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\rightarrow L+q\left({\dot {\mathbf {r} }}\cdot {\boldsymbol {\nabla }}+{\frac {\partial }{\partial t}}\right)f=L+q{\frac {df}{dt}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">→</mo> <mi>L</mi> <mo>+</mo> <mi>q</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>f</mi> <mo>=</mo> <mi>L</mi> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\rightarrow L+q\left({\dot {\mathbf {r} }}\cdot {\boldsymbol {\nabla }}+{\frac {\partial }{\partial t}}\right)f=L+q{\frac {df}{dt}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f158a747b38a73299c89cb7c11f3dc00858f1d41" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.634ex; height:6.176ex;" alt="{\displaystyle L\rightarrow L+q\left({\dot {\mathbf {r} }}\cdot {\boldsymbol {\nabla }}+{\frac {\partial }{\partial t}}\right)f=L+q{\frac {df}{dt}},}"> which still produces the same Lorentz force law. Note that the <a href="/wiki/Canonical_momentum" class="mw-redirect" title="Canonical momentum">canonical momentum</a> (conjugate to position r) is the <a href="/wiki/Kinetic_momentum" class="mw-redirect" title="Kinetic momentum">kinetic momentum</a> plus a contribution from the A field (known as the potential momentum): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} ={\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}=m{\dot {\mathbf {r} }}+q\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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} ={\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}=m{\dot {\mathbf {r} }}+q\mathbf {A} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b15411e72657747764e964991d9745121ea4ade" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.198ex; height:5.509ex;" alt="{\displaystyle \mathbf {p} ={\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}=m{\dot {\mathbf {r} }}+q\mathbf {A} .}"> This relation is also used in the <a href="/wiki/Minimal_coupling" title="Minimal coupling">minimal coupling</a> prescription in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> and <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. From this expression, we can see that the <a href="/wiki/Canonical_momentum" class="mw-redirect" title="Canonical momentum">canonical momentum</a> p is not gauge invariant, and therefore not a measurable physical quantity; However, if r is cyclic (i.e. Lagrangian is independent of position r), which happens if the ϕ and A fields are uniform, then this canonical momentum p given here is the conserved momentum, while the measurable physical kinetic momentum mv is not. <div class="mw-heading mw-heading2"><h2 id="Other_contexts_and_formulations">Other contexts and formulations</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=29" title="Edit section: Other contexts and formulations">edit</a>]</div> The ideas in Lagrangian mechanics have numerous applications in other areas of physics, and can adopt generalized results from the calculus of variations. <div class="mw-heading mw-heading3"><h3 id="Alternative_formulations_of_classical_mechanics">Alternative formulations of classical mechanics</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=30" title="Edit section: Alternative formulations of classical mechanics">edit</a>]</div> A closely related formulation of classical mechanics is <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>. The Hamiltonian is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28559600ff10ac7fe3da4578b37027e2b488f7df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.835ex; height:6.843ex;" alt="{\displaystyle H=\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-L}"> and can be obtained by performing a <a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre transformation</a> on the Lagrangian, which introduces new variables <a href="/wiki/Canonical_coordinates" title="Canonical coordinates">canonically conjugate</a> to the original variables. For example, given a set of generalized coordinates, the variables <a href="/wiki/Canonical_coordinates" title="Canonical coordinates">canonically conjugate</a> are the generalized momenta. This doubles the number of variables, but makes differential equations first order. The Hamiltonian is a particularly ubiquitous quantity in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> (see <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian (quantum mechanics)</a>). <a href="/wiki/Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a> is a hybrid formulation of Lagrangian and Hamiltonian mechanics, which is not often used in practice but an efficient formulation for cyclic coordinates. <div class="mw-heading mw-heading3"><h3 id="Momentum_space_formulation">Momentum space formulation</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=31" title="Edit section: Momentum space formulation">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/Position_and_momentum_space#Lagrangian_mechanics" class="mw-redirect" title="Position and momentum space">Position and momentum space § Lagrangian mechanics</a></div> The Euler–Lagrange equations can also be formulated in terms of the generalized momenta rather than generalized coordinates. Performing a Legendre transformation on the generalized coordinate Lagrangian L(q, dq/dt, t) obtains the generalized momenta Lagrangian L′(p, dp/dt, t) in terms of the original Lagrangian, as well the EL equations in terms of the generalized momenta. Both Lagrangians contain the same information, and either can be used to solve for the motion of the system. In practice generalized coordinates are more convenient to use and interpret than generalized momenta. <div class="mw-heading mw-heading3"><h3 id="Higher_derivatives_of_generalized_coordinates">Higher derivatives of generalized coordinates</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=32" title="Edit section: Higher derivatives of generalized coordinates">edit</a>]</div> There is no mathematical reason to restrict the derivatives of generalized coordinates to first order only. It is possible to derive modified EL equations for a Lagrangian containing higher order derivatives, see <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equation</a> for details. However, from the physical point-of-view there is an obstacle to include time derivatives higher than the first order, which is implied by Ostrogradsky's construction of a canonical formalism for nondegenerate higher derivative Lagrangians, see <a href="/wiki/Ostrogradsky_instability" title="Ostrogradsky instability">Ostrogradsky instability</a> <div class="mw-heading mw-heading3"><h3 id="Optics">Optics</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=33" title="Edit section: Optics">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/Hamiltonian_optics" title="Hamiltonian optics">Hamiltonian optics</a></div> Lagrangian mechanics can be applied to <a href="/wiki/Geometrical_optics" title="Geometrical optics">geometrical optics</a>, by applying variational principles to rays of light in a medium, and solving the EL equations gives the equations of the paths the light rays follow. <div class="mw-heading mw-heading3"><h3 id="Relativistic_formulation">Relativistic formulation</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=34" title="Edit section: Relativistic formulation">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/Relativistic_Lagrangian_mechanics" title="Relativistic Lagrangian mechanics">Relativistic Lagrangian mechanics</a></div> Lagrangian mechanics can be formulated in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> and <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. Some features of Lagrangian mechanics are retained in the relativistic theories but difficulties quickly appear in other respects. In particular, the EL equations take the same form, and the connection between cyclic coordinates and conserved momenta still applies, however the Lagrangian must be modified and is not simply the kinetic minus the potential energy of a particle. Also, it is not straightforward to handle multiparticle systems in a <a href="/wiki/Manifestly_covariant" class="mw-redirect" title="Manifestly covariant">manifestly covariant</a> way, it may be possible if a particular frame of reference is singled out. <div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=35" title="Edit section: Quantum mechanics">edit</a>]</div> In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <a href="/wiki/Action_(physics)" title="Action (physics)">action</a> and quantum-mechanical <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> are related via the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>, and the <a href="/wiki/Principle_of_stationary_action" class="mw-redirect" title="Principle of stationary action">principle of stationary action</a> can be understood in terms of <a href="/wiki/Constructive_interference" class="mw-redirect" title="Constructive interference">constructive interference</a> of <a href="/wiki/Wave_function" title="Wave function">wave functions</a>. In 1948, <a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a> discovered the <a href="/wiki/Path_integral_formulation" title="Path integral formulation">path integral formulation</a> extending the <a href="/wiki/Principle_of_least_action" class="mw-redirect" title="Principle of least action">principle of least action</a> to <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> for <a href="/wiki/Electrons" class="mw-redirect" title="Electrons">electrons</a> and <a href="/wiki/Photons" class="mw-redirect" title="Photons">photons</a>. In this formulation, particles travel every possible path between the initial and final states; the probability of a specific final state is obtained by summing over all possible trajectories leading to it. In the classical regime, the path integral formulation cleanly reproduces Hamilton's principle, and <a href="/wiki/Fermat%27s_principle" title="Fermat's principle">Fermat's principle</a> in <a href="/wiki/Optics" title="Optics">optics</a>. <div class="mw-heading mw-heading3"><h3 id="Classical_field_theory">Classical field theory</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=36" title="Edit section: Classical field theory">edit</a>]</div> In Lagrangian mechanics, the generalized coordinates form a discrete set of variables that define the configuration of a system. In <a href="/wiki/Classical_field_theory" title="Classical field theory">classical field theory</a>, the physical system is not a set of discrete particles, but rather a continuous field ϕ(r, t) defined over a region of 3D space. Associated with the field is a <a href="/wiki/Lagrangian_density" class="mw-redirect" title="Lagrangian density">Lagrangian density</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}(\phi ,\nabla \phi ,{\dot {\phi }},\mathbf {r} ,t)}"> <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> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>ϕ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</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 {\mathcal {L}}(\phi ,\nabla \phi ,{\dot {\phi }},\mathbf {r} ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6601834bf2cde4c5243491cc73f9ad49ad99c413" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.665ex; height:3.176ex;" alt="{\displaystyle {\mathcal {L}}(\phi ,\nabla \phi ,{\dot {\phi }},\mathbf {r} ,t)}"> defined in terms of the field and its space and time derivatives at a location r and time t. Analogous to the particle case, for non-relativistic applications the Lagrangian density is also the kinetic energy density of the field, minus its potential energy density (this is not true in general, and the Lagrangian density has to be "reverse engineered"). The Lagrangian is then the <a href="/wiki/Volume_integral" title="Volume integral">volume integral</a> of the Lagrangian density over 3D space <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(t)=\int {\mathcal {L}}\,\mathrm {d} ^{3}\mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(t)=\int {\mathcal {L}}\,\mathrm {d} ^{3}\mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6461d57eaa71f1da54957d342f3b3546583f6fb7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.351ex; height:5.676ex;" alt="{\displaystyle L(t)=\int {\mathcal {L}}\,\mathrm {d} ^{3}\mathbf {r} }"> where d3r is a 3D <a href="/wiki/Total_differential#Differentials_in_several_variables" class="mw-redirect" title="Total differential">differential</a> <a href="/wiki/Volume_element" title="Volume element">volume element</a>. The Lagrangian is a function of time since the Lagrangian density has implicit space dependence via the fields, and may have explicit spatial dependence, but these are removed in the integral, leaving only time in as the variable for the Lagrangian. <div class="mw-heading mw-heading3"><h3 id="Noether's_theorem">Noether's theorem</h3>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=37" title="Edit section: Noether's theorem">edit</a>]</div> The action principle, and the Lagrangian formalism, are tied closely to <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a>, which connects physical <a href="/wiki/Conserved_quantity" title="Conserved quantity">conserved quantities</a> to continuous <a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">symmetries</a> of a physical system. If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This characteristic is very helpful in showing that theories are consistent with either <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> or <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=38" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/28px-Crab_Nebula.jpg" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/42px-Crab_Nebula.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/56px-Crab_Nebula.jpg 2x" data-file-width="3864" data-file-height="3864" /><a href="/wiki/Portal:Astronomy" title="Portal:Astronomy">Astronomy portal</a></li></ul> <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 div-col-small" style="column-width: 20em;"> <ul><li><a href="/wiki/Canonical_coordinates" title="Canonical coordinates">Canonical coordinates</a></li> <li><a href="/wiki/Fundamental_lemma_of_the_calculus_of_variations" title="Fundamental lemma of the calculus of variations">Fundamental lemma of the calculus of variations</a></li> <li><a href="/wiki/Functional_derivative" title="Functional derivative">Functional derivative</a></li> <li><a href="/wiki/Generalized_coordinates" title="Generalized coordinates">Generalized coordinates</a></li> <li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li> <li><a href="/wiki/Hamiltonian_optics" title="Hamiltonian optics">Hamiltonian optics</a></li> <li><a href="/wiki/Inverse_problem_for_Lagrangian_mechanics" title="Inverse problem for Lagrangian mechanics">Inverse problem for Lagrangian mechanics</a>, the general topic of finding a Lagrangian for a system given the equations of motion.</li> <li><a href="/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field" title="Lagrangian and Eulerian specification of the flow field">Lagrangian and Eulerian specification of the flow field</a></li> <li><a href="/wiki/Lagrangian_point" class="mw-redirect" title="Lagrangian point">Lagrangian point</a></li> <li><a href="/wiki/Lagrangian_system" title="Lagrangian system">Lagrangian system</a></li> <li><a href="/wiki/Non-autonomous_mechanics" title="Non-autonomous mechanics">Non-autonomous mechanics</a></li> <li><a href="/wiki/Plateau%27s_problem" title="Plateau's problem">Plateau's problem</a></li> <li><a href="/wiki/Restricted_three-body_problem" class="mw-redirect" title="Restricted three-body problem">Restricted three-body problem</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=39" title="Edit section: Footnotes">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"><ol class="references"> <li id="cite_note-15"><a href="#cite_ref-15">^</a> Sometimes in this context the <a href="/wiki/Variational_derivative" class="mw-redirect" title="Variational derivative">variational derivative</a> denoted and defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta }{\delta \mathbf {r} _{k}}}\equiv {\frac {\partial }{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta }{\delta \mathbf {r} _{k}}}\equiv {\frac {\partial }{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf017cf1adcb1c2d424791e0c4385cc3f4be09fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.733ex; height:5.843ex;" alt="{\displaystyle {\frac {\delta }{\delta \mathbf {r} _{k}}}\equiv {\frac {\partial }{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}}"> is used. Throughout this article only partial and total derivatives are used. </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> Here the virtual displacements are assumed reversible, it is possible for some systems to have non-reversible virtual displacements that violate this principle, see <a href="/wiki/Udwadia%E2%80%93Kalaba_equation" class="mw-redirect" title="Udwadia–Kalaba equation">Udwadia–Kalaba equation</a>. </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> In other words <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042c0f4c6d0f69cd8b89ae11c6ed6090682cc21b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.199ex; height:2.676ex;" alt="{\displaystyle \mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0}"> for particle k subject to a constraint force, however <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{k\,x}\delta x_{k}\neq 0,\quad C_{k\,y}\delta y_{k}\neq 0,\quad C_{k\,z}\delta z_{k}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> </msub> <mi>δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mspace width="thinmathspace" /> <mi>y</mi> </mrow> </msub> <mi>δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mspace width="thinmathspace" /> <mi>z</mi> </mrow> </msub> <mi>δ</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{k\,x}\delta x_{k}\neq 0,\quad C_{k\,y}\delta y_{k}\neq 0,\quad C_{k\,z}\delta z_{k}\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c772d0de1f87babe632d9a1bdf70e05ba48d67a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.398ex; height:3.009ex;" alt="{\displaystyle C_{k\,x}\delta x_{k}\neq 0,\quad C_{k\,y}\delta y_{k}\neq 0,\quad C_{k\,z}\delta z_{k}\neq 0}"> because of the constraint equations on the rk coordinates. </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> The Lagrangian also can be written explicitly for a rotating frame. See Padmanabhan, 2000. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=40" title="Edit section: Notes">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> Fraser, Craig. "J. L. Lagrange's Early Contributions to the Principles and Methods of Mechanics". Archive for History of Exact Sciences, vol. 28, no. 3, 1983, pp. 197–241. JSTOR, <a rel="nofollow" class="external free" href="http://www.jstor.org/stable/41133689">http://www.jstor.org/stable/41133689</a>. 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American Journal of Physics. 73 (7): 603–610. <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/2005AmJPh..73..603H">2005AmJPh..73..603H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.1848516">10.1119/1.1848516</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9505">0002-9505</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Variational+mechanics+in+one+and+two+dimensions&rft.volume=73&rft.issue=7&rft.pages=603-610&rft.date=2005-07-01&rft.issn=0002-9505&rft_id=info%3Adoi%2F10.1119%2F1.1848516&rft_id=info%3Abibcode%2F2005AmJPh..73..603H&rft.aulast=Hanc&rft.aufirst=Jozef&rft.au=Taylor%2C+Edwin+F.&rft.au=Tuleja%2C+Slavomir&rft_id=https%3A%2F%2Fpubs.aip.org%2Fajp%2Farticle%2F73%2F7%2F603%2F1056166%2FVariational-mechanics-in-one-and-two-dimensions&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"> </li> <li id="cite_note-Hand_1998_page=44–45-38">^ <a href="#cite_ref-Hand_1998_page=44–45_38-0">a</a> <a href="#cite_ref-Hand_1998_page=44–45_38-1">b</a> <a href="#CITEREFHandFinch1998">Hand & Finch 1998</a>, p. 44–45 </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <a href="#CITEREFGoldstein1980">Goldstein 1980</a> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <a href="#CITEREFFetterWalecka1980">Fetter & Walecka 1980</a>, pp. 68–70 </li> <li id="cite_note-Landau_1976_page=4-41">^ <a href="#cite_ref-Landau_1976_page=4_41-0">a</a> <a href="#cite_ref-Landau_1976_page=4_41-1">b</a> <a href="#CITEREFLandauLifshitz1976">Landau & Lifshitz 1976</a>, p. 4 </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <a href="#CITEREFGoldsteinPooleSafko2002">Goldstein, Poole & Safko 2002</a>, p. 21 </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <a href="#CITEREFLandauLifshitz1976">Landau & Lifshitz 1976</a>, p. 4 </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <a href="#CITEREFGoldstein1980">Goldstein 1980</a>, p. 21 </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <a href="#CITEREFLandauLifshitz1976">Landau & Lifshitz 1976</a>, p. 14 </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <a href="#CITEREFLandauLifshitz1976">Landau & Lifshitz 1976</a>, p. 22 </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotheRothe2010" class="citation book cs1">Rothe, Heinz J; Rothe, Klaus D (2010). <a rel="nofollow" class="external text" href="https://www.worldscientific.com/worldscibooks/10.1142/7689">Classical and Quantum Dynamics of Constrained Hamiltonian Systems</a>. World Scientific Lecture Notes in Physics. Vol. 81. WORLD SCIENTIFIC. p. 7. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F7689">10.1142/7689</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4299-64-0" title="Special:BookSources/978-981-4299-64-0"><bdi>978-981-4299-64-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+and+Quantum+Dynamics+of+Constrained+Hamiltonian+Systems&rft.series=World+Scientific+Lecture+Notes+in+Physics&rft.pages=7&rft.pub=WORLD+SCIENTIFIC&rft.date=2010&rft_id=info%3Adoi%2F10.1142%2F7689&rft.isbn=978-981-4299-64-0&rft.aulast=Rothe&rft.aufirst=Heinz+J&rft.au=Rothe%2C+Klaus+D&rft_id=https%3A%2F%2Fwww.worldscientific.com%2Fworldscibooks%2F10.1142%2F7689&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <a href="#CITEREFTaylor2005">Taylor 2005</a>, p. 297 </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <a href="#CITEREFPadmanabhan2000">Padmanabhan 2000</a>, p. 48 </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <a href="#CITEREFHandFinch1998">Hand & Finch 1998</a>, pp. 140–141 </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <a href="#CITEREFHildebrand1992">Hildebrand 1992</a>, p. 156 </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <a href="#CITEREFZakZbilutMeyers1997">Zak, Zbilut & Meyers 1997</a>, pp. 202 </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> <a href="#CITEREFShabana2008">Shabana 2008</a>, pp. 118–119 </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <a href="#CITEREFGannon2006">Gannon 2006</a>, p. 267 </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <a href="#CITEREFKosyakov2007">Kosyakov 2007</a> </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <a href="#CITEREFGalley2013">Galley 2013</a> </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <a href="#CITEREFBirnholtzHadarKol2014">Birnholtz, Hadar & Kol 2014</a> </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> <a href="#CITEREFBirnholtzHadarKol2013">Birnholtz, Hadar & Kol 2013</a> </li> <li id="cite_note-Torby_1984_page=271-60">^ <a href="#cite_ref-Torby_1984_page=271_60-0">a</a> <a href="#cite_ref-Torby_1984_page=271_60-1">b</a> <a href="#CITEREFTorby1984">Torby 1984</a>, p. 271 </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=41" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLagrange1811" class="citation book cs1">Lagrange, J. 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Vol. 2.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=M%C3%A9canique+analytique&rft.date=1815&rft.aulast=Lagrange&rft.aufirst=J.+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTmMSAAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPenrose2007" class="citation book cs1">Penrose, Roger (2007). The Road to Reality. Vintage books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-679-77631-4" title="Special:BookSources/978-0-679-77631-4"><bdi>978-0-679-77631-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Road+to+Reality&rft.pub=Vintage+books&rft.date=2007&rft.isbn=978-0-679-77631-4&rft.aulast=Penrose&rft.aufirst=Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz1976" class="citation book cs1"><a href="/wiki/L._D._Landau" class="mw-redirect" title="L. D. Landau">Landau, L. D.</a>; <a href="/wiki/E._M._Lifshitz" class="mw-redirect" title="E. M. Lifshitz">Lifshitz, E. M.</a> (15 January 1976). Mechanics (3rd ed.). Butterworth Heinemann. p. 134. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780750628969" title="Special:BookSources/9780750628969"><bdi>9780750628969</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics&rft.pages=134&rft.edition=3rd&rft.pub=Butterworth+Heinemann&rft.date=1976-01-15&rft.isbn=9780750628969&rft.aulast=Landau&rft.aufirst=L.+D.&rft.au=Lifshitz%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz1975" class="citation book cs1"><a href="/wiki/Lev_Davidovich_Landau" class="mw-redirect" title="Lev Davidovich Landau">Landau, Lev</a>; <a href="/wiki/Evgeny_Mikhailovich_Lifshitz" class="mw-redirect" title="Evgeny Mikhailovich Lifshitz">Lifshitz, Evgeny</a> (1975). The Classical Theory of Fields. Elsevier Ltd. <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.pub=Elsevier+Ltd.&rft.date=1975&rft.isbn=978-0-7506-2768-9&rft.aulast=Landau&rft.aufirst=Lev&rft.au=Lifshitz%2C+Evgeny&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHandFinch1998" class="citation book cs1">Hand, L. N.; Finch, J. D. (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA141">Analytical Mechanics</a> (2nd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521575720" title="Special:BookSources/9780521575720"><bdi>9780521575720</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytical+Mechanics&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1998&rft.isbn=9780521575720&rft.aulast=Hand&rft.aufirst=L.+N.&rft.au=Finch%2C+J.+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1J2hzvX2Xh8C%26pg%3DPA141&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoséSaletan1998" class="citation book cs1">Saletan, E. J.; José, J. V. (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Eql9dRQDgvQC&pg=PA129">Classical Dynamics: A Contemporary Approach</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521636360" title="Special:BookSources/9780521636360"><bdi>9780521636360</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+Dynamics%3A+A+Contemporary+Approach&rft.pub=Cambridge+University+Press&rft.date=1998&rft.isbn=9780521636360&rft.aulast=Saletan&rft.aufirst=E.+J.&rft.au=Jos%C3%A9%2C+J.+V.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEql9dRQDgvQC%26pg%3DPA129&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKibbleBerkshire2004" class="citation book cs1">Kibble, T. W. B.; Berkshire, F. H. (2004). Classical Mechanics (5th ed.). Imperial College Press. p. 236. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781860944352" title="Special:BookSources/9781860944352"><bdi>9781860944352</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+Mechanics&rft.pages=236&rft.edition=5th&rft.pub=Imperial+College+Press&rft.date=2004&rft.isbn=9781860944352&rft.aulast=Kibble&rft.aufirst=T.+W.+B.&rft.au=Berkshire%2C+F.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldstein1980" class="citation book cs1"><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Goldstein, Herbert</a> (1980). Classical Mechanics (2nd ed.). San Francisco, CA: Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0201029189" title="Special:BookSources/0201029189"><bdi>0201029189</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+Mechanics&rft.place=San+Francisco%2C+CA&rft.edition=2nd&rft.pub=Addison+Wesley&rft.date=1980&rft.isbn=0201029189&rft.aulast=Goldstein&rft.aufirst=Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldsteinPooleSafko2002" class="citation book cs1"><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Goldstein, Herbert</a>; <a href="/w/index.php?title=Charles_P._Poole&action=edit&redlink=1" class="new" title="Charles P. Poole (page does not exist)">Poole, Charles P. Jr.</a>; Safko, John L. (2002). <a rel="nofollow" class="external text" href="https://archive.org/details/classicalmechani00gold">Classical Mechanics</a> (3rd ed.). San Francisco, CA: Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-65702-3" title="Special:BookSources/0-201-65702-3"><bdi>0-201-65702-3</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+Mechanics&rft.place=San+Francisco%2C+CA&rft.edition=3rd&rft.pub=Addison+Wesley&rft.date=2002&rft.isbn=0-201-65702-3&rft.aulast=Goldstein&rft.aufirst=Herbert&rft.au=Poole%2C+Charles+P.+Jr.&rft.au=Safko%2C+John+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fclassicalmechani00gold&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLanczos1986" class="citation book cs1">Lanczos, Cornelius (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PA43">"II §5 Auxiliary conditions: the Lagrangian λ-method"</a>. 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"Theory of post-Newtonian radiation and reaction". Physical Review D. 88 (10): 104037. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1305.6930">1305.6930</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/2013PhRvD..88j4037B">2013PhRvD..88j4037B</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%2FPhysRevD.88.104037">10.1103/PhysRevD.88.104037</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:119170985">119170985</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Theory+of+post-Newtonian+radiation+and+reaction&rft.volume=88&rft.issue=10&rft.pages=104037&rft.date=2013&rft_id=info%3Aarxiv%2F1305.6930&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119170985%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.88.104037&rft_id=info%3Abibcode%2F2013PhRvD..88j4037B&rft.aulast=Birnholtz&rft.aufirst=Ofek&rft.au=Hadar%2C+Shahar&rft.au=Kol%2C+Barak&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoger_F_Gans2013" class="citation book cs1">Roger F Gans (2013). <a rel="nofollow" class="external text" href="https://www.amazon.com/gp/product/1461439299/ref=olp_product_details?ie=UTF8&me=&seller=">Engineering Dynamics: From the Lagrangian to Simulation</a>. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-3929-5" title="Special:BookSources/978-1-4614-3929-5"><bdi>978-1-4614-3929-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Dynamics%3A+From+the+Lagrangian+to+Simulation&rft.place=New+York&rft.pub=Springer&rft.date=2013&rft.isbn=978-1-4614-3929-5&rft.au=Roger+F+Gans&rft_id=https%3A%2F%2Fwww.amazon.com%2Fgp%2Fproduct%2F1461439299%2Fref%3Dolp_product_details%3Fie%3DUTF8%26me%3D%26seller%3D&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGannon2006" class="citation book cs1">Gannon, Terry (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ehrUt21SnsoC&pg=RA3-PA267">Moonshine beyond the monster: the bridge connecting algebra, modular forms and physics</a>. Cambridge University Press. p. 267. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-83531-3" title="Special:BookSources/0-521-83531-3"><bdi>0-521-83531-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Moonshine+beyond+the+monster%3A+the+bridge+connecting+algebra%2C+modular+forms+and+physics&rft.pages=267&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=0-521-83531-3&rft.aulast=Gannon&rft.aufirst=Terry&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DehrUt21SnsoC%26pg%3DRA3-PA267&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTorby1984" class="citation book cs1">Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-063366-4" title="Special:BookSources/0-03-063366-4"><bdi>0-03-063366-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Energy+Methods&rft.btitle=Advanced+Dynamics+for+Engineers&rft.place=United+States+of+America&rft.series=HRW+Series+in+Mechanical+Engineering&rft.pub=CBS+College+Publishing&rft.date=1984&rft.isbn=0-03-063366-4&rft.aulast=Torby&rft.aufirst=Bruce&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFosterNightingale1995" class="citation book cs1">Foster, J; Nightingale, J.D. (1995). A Short Course in General Relativity (2nd ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-063366-4" title="Special:BookSources/0-03-063366-4"><bdi>0-03-063366-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Short+Course+in+General+Relativity&rft.edition=2nd&rft.pub=Springer&rft.date=1995&rft.isbn=0-03-063366-4&rft.aulast=Foster&rft.aufirst=J&rft.au=Nightingale%2C+J.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFM._P._HobsonG._P._EfstathiouA._N._Lasenby2006" class="citation book cs1">M. P. Hobson; G. P. Efstathiou; A. N. Lasenby (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xma1QuTJphYC&q=hobson+general+relativity+gravitomagnetic+field&pg=PA496">General Relativity: An Introduction for Physicists</a>. Cambridge University Press. pp. 79–80. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521829519" title="Special:BookSources/9780521829519"><bdi>9780521829519</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Relativity%3A+An+Introduction+for+Physicists&rft.pages=79-80&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=9780521829519&rft.au=M.+P.+Hobson&rft.au=G.+P.+Efstathiou&rft.au=A.+N.+Lasenby&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dxma1QuTJphYC%26q%3Dhobson%2Bgeneral%2Brelativity%2Bgravitomagnetic%2Bfield%26pg%3DPA496&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSyngeSchild1949" class="citation book cs1">Synge, J.L.; Schild, A. (1949). <a rel="nofollow" class="external text" href="https://archive.org/details/tensorcalculus00syng">Tensor Calculus</a>. first Dover Publications 1978 edition. <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"><bdi>978-0-486-63612-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tensor+Calculus&rft.pub=first+Dover+Publications+1978+edition&rft.date=1949&rft.isbn=978-0-486-63612-2&rft.aulast=Synge&rft.aufirst=J.L.&rft.au=Schild%2C+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftensorcalculus00syng&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKay1988" class="citation book cs1">Kay, David (April 1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6tUU3KruG14C">Schaum's Outline of Tensor Calculus</a>. McGraw Hill Professional. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-033484-7" title="Special:BookSources/978-0-07-033484-7"><bdi>978-0-07-033484-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schaum%27s+Outline+of+Tensor+Calculus&rft.pub=McGraw+Hill+Professional&rft.date=1988-04&rft.isbn=978-0-07-033484-7&rft.aulast=Kay&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6tUU3KruG14C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=42" title="Edit section: Further reading">edit</a>]</div> <ul><li>Gupta, Kiran Chandra, Classical mechanics of particles and rigid bodies (Wiley, 1988).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCassel2013" class="citation book cs1">Cassel, Kevin (2013). Variational methods with applications in science and engineering. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-02258-4" title="Special:BookSources/978-1-107-02258-4"><bdi>978-1-107-02258-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Variational+methods+with+applications+in+science+and+engineering&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-1-107-02258-4&rft.aulast=Cassel&rft.aufirst=Kevin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Goldstein</a>, Herbert, et al. <a href="/wiki/Classical_Mechanics_(Goldstein_book)" class="mw-redirect" title="Classical Mechanics (Goldstein book)">Classical Mechanics</a>. 3rd ed., Pearson, 2002.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Lagrangian_mechanics&action=edit&section=43" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDAMTP" class="citation web cs1">David Tong. <a rel="nofollow" class="external text" href="http://www.damtp.cam.ac.uk/user/tong/dynamics.html">"Cambridge Lecture Notes on Classical Dynamics"</a>. DAMTP. Retrieved 2017-06-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=DAMTP&rft.atitle=Cambridge+Lecture+Notes+on+Classical+Dynamics&rft.au=David+Tong&rft_id=http%3A%2F%2Fwww.damtp.cam.ac.uk%2Fuser%2Ftong%2Fdynamics.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+mechanics" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.eftaylor.com/software/ActionApplets/LeastAction.html">Principle of least action interactive</a> Excellent interactive explanation/webpage</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130513140933/http://portail.mathdoc.fr/cgi-bin/oetoc?id=OE_LAGRANGE__1">Joseph Louis de Lagrange - Œuvres complètes</a> (Gallica-Math)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170918105452/http://www.physics.indiana.edu/~dermisek/CM_13/CM-7-2p.pdf">Constrained motion and generalized coordinates</a>, page 4</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output 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href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Computational_statistics" title="Computational statistics">Statistics</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Mathematical_software" scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Mathematical_software" title="Mathematical software">Mathematical software</a></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/List_of_arbitrary-precision_arithmetic_software" title="List of arbitrary-precision arithmetic software">Arbitrary-precision arithmetic</a></li> <li><a href="/wiki/List_of_finite_element_software_packages" title="List of finite element software packages">Finite element analysis</a></li> <li><a href="/wiki/Tensor_software" title="Tensor software">Tensor software</a></li> <li><a href="/wiki/List_of_interactive_geometry_software" title="List of interactive geometry software">Interactive geometry software</a></li> <li><a href="/wiki/List_of_optimization_software" title="List of optimization software">Optimization software</a></li> <li><a href="/wiki/List_of_statistical_software" title="List of statistical software">Statistical software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical-analysis software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical libraries</a></li> <li><a href="/wiki/Solver" title="Solver">Solvers</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/Discrete_mathematics" title="Discrete mathematics">Discrete</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/Computer_algebra" title="Computer algebra">Computer algebra</a></li> <li><a href="/wiki/Computational_number_theory" title="Computational number theory">Computational number theory</a></li> <li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></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/Approximation_theory" title="Approximation theory">Approximation theory</a></li> <li><a href="/wiki/Clifford_analysis" title="Clifford analysis">Clifford analysis</a> <ul><li><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equations</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equations</a></li></ul></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a> <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/Geometric_analysis" title="Geometric analysis">Geometric analysis</a></li></ul></li> <li><a href="/wiki/Dynamical_system" title="Dynamical system">Dynamical systems</a> <ul><li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li></ul></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> <ul><li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a></li></ul></li> <li><a href="/wiki/Harmonic_analysis_(mathematics)" class="mw-redirect" title="Harmonic analysis (mathematics)">Harmonic analysis</a> <ul><li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li></ul></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a> <ul><li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Vector_calculus#Vector_algebra" title="Vector calculus">Vector</a></li></ul></li> <li><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a> <ul><li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></li></ul></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> <ul><li><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical linear algebra</a></li> <li><a href="/wiki/Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li> <li><a href="/wiki/Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Variational calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</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/Probability_distribution" title="Probability distribution">Distributions</a> (<a href="/wiki/Random_variable" title="Random variable">random variables</a>)</li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic processes</a> / <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">analysis</a></li> <li><a href="/wiki/Functional_integration" title="Functional integration">Path integral</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Stochastic variational calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></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/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a> <ul><li><a class="mw-selflink selflink">Lagrangian</a></li> <li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a></li></ul></li> <li><a href="/wiki/Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">Field theory</a> <ul><li><a href="/wiki/Classical_field_theory" title="Classical field theory">Classical</a></li> <li><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal</a></li> <li><a href="/wiki/Effective_field_theory" title="Effective field theory">Effective</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum</a></li> <li><a href="/wiki/Statistical_field_theory" title="Statistical field theory">Statistical</a></li> <li><a href="/wiki/Topological_field_theory" class="mw-redirect" title="Topological field theory">Topological</a></li></ul></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a> <ul><li><a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">in quantum mechanics</a></li></ul></li> <li><a href="/wiki/Potential_theory" title="Potential theory">Potential theory</a></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a> <ul><li><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic</a></li> <li><a href="/wiki/Topological_string_theory" title="Topological string theory">Topological</a></li></ul></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a> <ul><li><a href="/wiki/Supersymmetric_quantum_mechanics" title="Supersymmetric quantum mechanics">Supersymmetric quantum mechanics</a></li> <li><a href="/wiki/Supersymmetric_theory_of_stochastic_dynamics" title="Supersymmetric theory of stochastic dynamics">Supersymmetric theory of stochastic dynamics</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Algebraic_structures" scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Algebraic_structures" class="mw-redirect" title="Algebraic structures">Algebraic structures</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/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Feynman integral</a></li> <li><a href="/wiki/Poisson_algebra" title="Poisson algebra">Poisson algebra</a></li> <li><a href="/wiki/Quantum_group" title="Quantum group">Quantum group</a></li> <li><a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group</a></li> <li><a href="/wiki/Particle_physics_and_representation_theory" title="Particle physics and representation theory">Representation theory</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Superalgebra" title="Superalgebra">Superalgebra</a></li> <li><a href="/wiki/Supersymmetry_algebra" title="Supersymmetry algebra">Supersymmetry algebra</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/Decision_theory" title="Decision theory">Decision sciences</a></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/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice theory</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other applications</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/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Sociology</a></li> <li>"<a href="/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" title="The Unreasonable Effectiveness of Mathematics in the Natural Sciences">The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a>"</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/Mathematics" title="Mathematics">Mathematics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</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/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a> <ul><li><a href="/wiki/Japan_Society_for_Industrial_and_Applied_Mathematics" title="Japan Society for Industrial and Applied Mathematics">Japan Society for Industrial and Applied Mathematics</a></li></ul></li> <li><a href="/wiki/Soci%C3%A9t%C3%A9_de_Math%C3%A9matiques_Appliqu%C3%A9es_et_Industrielles" title="Société de Mathématiques Appliquées et Industrielles">Société de Mathématiques Appliquées et Industrielles</a></li> <li><a href="/wiki/International_Council_for_Industrial_and_Applied_Mathematics" title="International Council for Industrial and Applied Mathematics">International Council for Industrial and Applied Mathematics</a></li> <li><a href="/w/index.php?title=European_Community_on_Computational_Methods_in_Applied_Sciences&action=edit&redlink=1" class="new" title="European Community on Computational Methods in Applied Sciences (page does not exist)">European Community on Computational Methods in Applied Sciences</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Category:Mathematics" title="Category:Mathematics">Category</a></li> <li><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a> / <a href="/wiki/Topic_outline_of_mathematics" class="mw-redirect" title="Topic outline of mathematics">outline</a> / <a href="/wiki/List_of_mathematics_topics" class="mw-redirect" title="List of mathematics topics">topics list</a></li></ul> </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="Major_branches_of_physics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse 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:Branches_of_physics" title="Template:Branches of physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Branches_of_physics" title="Template talk:Branches of physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Branches_of_physics" title="Special:EditPage/Template:Branches of physics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_branches_of_physics" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Branches_of_physics" title="Branches of physics">branches of physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Basic_research" title="Basic research">Pure</a></li> <li><a href="/wiki/Applied_physics" title="Applied physics">Applied</a> <ul><li><a href="/wiki/Engineering_physics" title="Engineering physics">Engineering</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Approaches</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Experimental_physics" title="Experimental physics">Experimental</a></li> <li><a href="/wiki/Theoretical_physics" title="Theoretical physics">Theoretical</a> <ul><li><a href="/wiki/Computational_physics" title="Computational physics">Computational</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Classical_physics" title="Classical physics">Classical</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a> <ul><li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newtonian</a></li> <li><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical</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></ul></li> <li><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></li> <li><a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">Classical electromagnetism</a></li> <li><a href="/wiki/Classical_optics" class="mw-redirect" title="Classical optics">Classical optics</a> <ul><li><a href="/wiki/Geometrical_optics" title="Geometrical optics">Ray</a></li> <li><a href="/wiki/Physical_optics" title="Physical optics">Wave</a></li></ul></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a> <ul><li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Modern_physics" title="Modern physics">Modern</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Relativistic_mechanics" title="Relativistic mechanics">Relativistic mechanics</a> <ul><li><a href="/wiki/Special_relativity" title="Special relativity">Special</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General</a></li></ul></li> <li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li> <li><a href="/wiki/Particle_physics" title="Particle physics">Particle physics</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Atomic,_molecular,_and_optical_physics" title="Atomic, molecular, and optical physics">Atomic, molecular, and optical physics</a> <ul><li><a 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style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_physics" title="History of physics">History of physics</a></li> <li><a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a></li> <li><a href="/wiki/Philosophy_of_physics" title="Philosophy of physics">Philosophy of physics</a></li> <li><a href="/wiki/Physics_education" title="Physics education">Physics education</a></li> <li><a href="/wiki/Timeline_of_fundamental_physics_discoveries" title="Timeline of fundamental physics discoveries">Timeline of physics discoveries</a></li></ul> </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 authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" 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Lagrangian mechanics - Wikipedia