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href="#Harmonic_motion"> <div class="vector-toc-text"> 2.5.3 Harmonic motion </div> </a> <ul id="toc-Harmonic_motion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Objects_with_variable_mass" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Objects_with_variable_mass"> <div class="vector-toc-text"> 2.5.4 Objects with variable mass </div> </a> <ul id="toc-Objects_with_variable_mass-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Work_and_energy" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Work_and_energy"> <div class="vector-toc-text"> 3 Work and energy </div> </a> <ul id="toc-Work_and_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rigid-body_motion_and_rotation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Rigid-body_motion_and_rotation"> <div class="vector-toc-text"> 4 Rigid-body motion and rotation </div> </a> <button aria-controls="toc-Rigid-body_motion_and_rotation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Rigid-body motion and rotation subsection </button> <ul id="toc-Rigid-body_motion_and_rotation-sublist" class="vector-toc-list"> <li id="toc-Center_of_mass" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Center_of_mass"> <div class="vector-toc-text"> 4.1 Center of mass </div> </a> <ul id="toc-Center_of_mass-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotational_analogues_of_Newton's_laws" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rotational_analogues_of_Newton's_laws"> <div class="vector-toc-text"> 4.2 Rotational analogues of Newton's laws </div> </a> <ul id="toc-Rotational_analogues_of_Newton's_laws-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multi-body_gravitational_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multi-body_gravitational_system"> <div class="vector-toc-text"> 4.3 Multi-body gravitational system </div> </a> <ul id="toc-Multi-body_gravitational_system-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Chaos_and_unpredictability" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Chaos_and_unpredictability"> <div class="vector-toc-text"> 5 Chaos and unpredictability </div> </a> <button aria-controls="toc-Chaos_and_unpredictability-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Chaos and unpredictability subsection </button> <ul id="toc-Chaos_and_unpredictability-sublist" class="vector-toc-list"> <li id="toc-Nonlinear_dynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonlinear_dynamics"> <div class="vector-toc-text"> 5.1 Nonlinear dynamics </div> </a> <ul id="toc-Nonlinear_dynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Singularities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Singularities"> <div class="vector-toc-text"> 5.2 Singularities </div> </a> <ul id="toc-Singularities-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relation_to_other_formulations_of_classical_physics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_to_other_formulations_of_classical_physics"> <div class="vector-toc-text"> 6 Relation to other formulations of classical physics </div> </a> <button aria-controls="toc-Relation_to_other_formulations_of_classical_physics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Relation to other formulations of classical physics subsection </button> <ul id="toc-Relation_to_other_formulations_of_classical_physics-sublist" class="vector-toc-list"> <li id="toc-Lagrangian" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lagrangian"> <div class="vector-toc-text"> 6.1 Lagrangian </div> </a> <ul id="toc-Lagrangian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hamiltonian" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hamiltonian"> <div class="vector-toc-text"> 6.2 Hamiltonian </div> </a> <ul id="toc-Hamiltonian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hamilton–Jacobi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hamilton–Jacobi"> <div class="vector-toc-text"> 6.3 Hamilton–Jacobi </div> </a> <ul id="toc-Hamilton–Jacobi-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relation_to_other_physical_theories" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_to_other_physical_theories"> <div class="vector-toc-text"> 7 Relation to other physical theories </div> </a> <button aria-controls="toc-Relation_to_other_physical_theories-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Relation to other physical theories subsection </button> <ul id="toc-Relation_to_other_physical_theories-sublist" class="vector-toc-list"> <li id="toc-Thermodynamics_and_statistical_physics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thermodynamics_and_statistical_physics"> <div class="vector-toc-text"> 7.1 Thermodynamics and statistical physics </div> </a> <ul id="toc-Thermodynamics_and_statistical_physics-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"> 7.2 Electromagnetism </div> </a> <ul id="toc-Electromagnetism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special_relativity"> <div class="vector-toc-text"> 7.3 Special relativity </div> </a> <ul id="toc-Special_relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_relativity"> <div class="vector-toc-text"> 7.4 General relativity </div> </a> <ul id="toc-General_relativity-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"> 7.5 Quantum mechanics </div> </a> <ul id="toc-Quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 8 History </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle History subsection </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Antiquity_and_medieval_background" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Antiquity_and_medieval_background"> <div class="vector-toc-text"> 8.1 Antiquity and medieval background </div> </a> <ul id="toc-Antiquity_and_medieval_background-sublist" class="vector-toc-list"> <li id="toc-Aristotle_and_"violent"_motion" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Aristotle_and_"violent"_motion"> <div class="vector-toc-text"> 8.1.1 Aristotle and "violent" motion </div> </a> <ul id="toc-Aristotle_and_"violent"_motion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Philoponus_and_impetus" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Philoponus_and_impetus"> <div class="vector-toc-text"> 8.1.2 Philoponus and impetus </div> </a> <ul id="toc-Philoponus_and_impetus-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Inertia_and_the_first_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inertia_and_the_first_law"> <div class="vector-toc-text"> 8.2 Inertia and the first law </div> </a> <ul id="toc-Inertia_and_the_first_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Force_and_the_second_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Force_and_the_second_law"> <div class="vector-toc-text"> 8.3 Force and the second law </div> </a> <ul id="toc-Force_and_the_second_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Momentum_conservation_and_the_third_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Momentum_conservation_and_the_third_law"> <div class="vector-toc-text"> 8.4 Momentum conservation and the third law </div> </a> <ul id="toc-Momentum_conservation_and_the_third_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-After_the_Principia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#After_the_Principia"> <div class="vector-toc-text"> 8.5 After the Principia </div> </a> <ul id="toc-After_the_Principia-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"> 9 See also </div> </a> <ul id="toc-See_also-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"> 10 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"> 11 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"> 12 Further reading </div> </a> <ul id="toc-Further_reading-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">Newton's laws of motion</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 116 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-116" aria-hidden="true" > 116 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/Newton_se_bewegingswette" title="Newton se bewegingswette – Afrikaans" lang="af" hreflang="af" data-title="Newton se bewegingswette" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Newtonsche_Gesetze" title="Newtonsche Gesetze – Alemannic" lang="gsw" hreflang="gsw" data-title="Newtonsche Gesetze" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D9%88%D8%A7%D9%86%D9%8A%D9%86_%D9%86%D9%8A%D9%88%D8%AA%D9%86_%D9%84%D9%84%D8%AD%D8%B1%D9%83%D8%A9" title="قوانين نيوتن للحركة – Arabic" lang="ar" hreflang="ar" data-title="قوانين نيوتن للحركة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%86%D5%AB%D6%82%D5%A9%D5%B8%D5%B6%D5%AB_%D5%B7%D5%A1%D6%80%D5%AA%D5%B4%D5%A1%D5%B6_%D6%85%D6%80%D5%A7%D5%B6%D6%84%D5%B6%D5%A5%D6%80" title="Նիւթոնի շարժման օրէնքներ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Նիւթոնի շարժման օրէնքներ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target">Արեւմտահայերէն</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%A8%E0%A6%BF%E0%A6%89%E0%A6%9F%E0%A6%A8%E0%A7%B0_%E0%A6%97%E0%A6%A4%E0%A6%BF%E0%A7%B0_%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A7%B0" title="নিউটনৰ গতিৰ সূত্ৰ – Assamese" lang="as" hreflang="as" data-title="নিউটনৰ গতিৰ সূত্ৰ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Lleis_de_Newton" title="Lleis de Newton – Asturian" lang="ast" hreflang="ast" data-title="Lleis de Newton" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Nyuton_qanunlar%C4%B1" title="Nyuton qanunları – Azerbaijani" lang="az" hreflang="az" data-title="Nyuton qanunları" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%86%DB%8C%D9%88%D8%AA%D9%88%D9%86%D9%88%D9%86_%D8%AD%D8%B1%DA%A9%D8%AA_%D9%82%D8%A7%D9%86%D9%88%D9%86%D9%84%D8%A7%D8%B1%DB%8C" title="نیوتونون حرکت قانونلاری – South Azerbaijani" lang="azb" hreflang="azb" data-title="نیوتونون حرکت قانونلاری" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-ban mw-list-item"><a href="https://ban.wikipedia.org/wiki/Hukum_gerak_Newton" title="Hukum gerak Newton – Balinese" lang="ban" hreflang="ban" data-title="Hukum gerak Newton" data-language-autonym="Basa Bali" data-language-local-name="Balinese" class="interlanguage-link-target">Basa Bali</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A8%E0%A6%BF%E0%A6%89%E0%A6%9F%E0%A6%A8%E0%A7%87%E0%A6%B0_%E0%A6%97%E0%A6%A4%E0%A6%BF%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%B8%E0%A6%AE%E0%A7%82%E0%A6%B9" title="নিউটনের গতিসূত্রসমূহ – Bangla" lang="bn" hreflang="bn" data-title="নিউটনের গতিসূত্রসমূহ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD%D1%8B_%D0%9D%D1%8C%D1%8E%D1%82%D0%B0%D0%BD%D0%B0" title="Законы Ньютана – Belarusian" lang="be" hreflang="be" data-title="Законы Ньютана" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD%D1%8B_%D0%9D%D1%8C%D1%8E%D1%82%D0%B0%D0%BD%D0%B0" title="Законы Ньютана – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Законы Ньютана" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A5%82%E0%A4%9F%E0%A4%A8_%E0%A4%95%E0%A5%87_%E0%A4%97%E0%A4%A4%E0%A4%BF_%E0%A4%95%E0%A5%87_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="न्यूटन के गति के नियम – Bhojpuri" lang="bh" hreflang="bh" data-title="न्यूटन के गति के नियम" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target">भोजपुरी</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8_%D0%BD%D0%B0_%D0%9D%D1%8E%D1%82%D0%BE%D0%BD" title="Закони на Нютон – Bulgarian" lang="bg" hreflang="bg" data-title="Закони на Нютон" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Newtonsche_Axiome" title="Newtonsche Axiome – Bavarian" lang="bar" hreflang="bar" data-title="Newtonsche Axiome" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target">Boarisch</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Newtonovi_zakoni_kretanja" title="Newtonovi zakoni kretanja – Bosnian" lang="bs" hreflang="bs" data-title="Newtonovi zakoni kretanja" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D0%BE%D0%B9_%D1%85%D1%83%D1%83%D0%BB%D0%B8%D0%BD%D1%83%D1%83%D0%B4" title="Ньютоной хуулинууд – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Ньютоной хуулинууд" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Lleis_de_Newton" title="Lleis de Newton – Catalan" lang="ca" hreflang="ca" data-title="Lleis de Newton" 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%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD_%D1%81%D0%B0%D0%BA%D0%BA%D1%83%D0%BD%C4%95%D1%81%D0%B5%D0%BC" title="Ньютон саккунĕсем – Chuvash" lang="cv" hreflang="cv" data-title="Ньютон саккунĕсем" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Newtonovy_pohybov%C3%A9_z%C3%A1kony" title="Newtonovy pohybové zákony – Czech" lang="cs" hreflang="cs" data-title="Newtonovy pohybové zákony" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Mitemo_yaNewton_paMuhambo" title="Mitemo yaNewton paMuhambo – Shona" lang="sn" hreflang="sn" data-title="Mitemo yaNewton paMuhambo" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Deddfau_mudiant_Newton" title="Deddfau mudiant Newton – Welsh" lang="cy" hreflang="cy" data-title="Deddfau mudiant Newton" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Newtons_love" title="Newtons love – Danish" lang="da" hreflang="da" data-title="Newtons love" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Newtonsche_Gesetze" title="Newtonsche Gesetze – German" lang="de" hreflang="de" data-title="Newtonsche Gesetze" 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/Newtoni_seadused" title="Newtoni seadused – Estonian" lang="et" hreflang="et" data-title="Newtoni seadused" 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%9D%CF%8C%CE%BC%CE%BF%CE%B9_%CE%BA%CE%AF%CE%BD%CE%B7%CF%83%CE%B7%CF%82_%CF%84%CE%BF%CF%85_%CE%9D%CE%B5%CF%8D%CF%84%CF%89%CE%BD%CE%B1" title="Νόμοι κίνησης του Νεύτωνα – Greek" lang="el" hreflang="el" data-title="Νόμοι κίνησης του Νεύτωνα" 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/Leyes_de_Newton" title="Leyes de Newton – Spanish" lang="es" hreflang="es" data-title="Leyes de Newton" 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/Le%C4%9Doj_de_Newton_pri_movo" title="Leĝoj de Newton pri movo – Esperanto" lang="eo" hreflang="eo" data-title="Leĝoj de Newton pri movo" 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/Newtonen_legeak" title="Newtonen legeak – Basque" lang="eu" hreflang="eu" data-title="Newtonen legeak" 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%82%D9%88%D8%A7%D9%86%DB%8C%D9%86_%D8%AD%D8%B1%DA%A9%D8%AA_%D9%86%DB%8C%D9%88%D8%AA%D9%86" title="قوانین حرکت نیوتن – Persian" lang="fa" hreflang="fa" data-title="قوانین حرکت نیوتن" 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/Lois_du_mouvement_de_Newton" title="Lois du mouvement de Newton – French" lang="fr" hreflang="fr" data-title="Lois du mouvement de Newton" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Wetten_fan_Newton" title="Wetten fan Newton – Western Frisian" lang="fy" hreflang="fy" data-title="Wetten fan Newton" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target">Frysk</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Dl%C3%ADthe_gluaisne_Newton" title="Dlíthe gluaisne Newton – Irish" lang="ga" hreflang="ga" data-title="Dlíthe gluaisne Newton" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Leis_de_Newton" title="Leis de Newton – Galician" lang="gl" hreflang="gl" data-title="Leis de Newton" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Mawatho_matano_ma_Newton" title="Mawatho matano ma Newton – Kikuyu" lang="ki" hreflang="ki" data-title="Mawatho matano ma Newton" data-language-autonym="Gĩkũyũ" data-language-local-name="Kikuyu" class="interlanguage-link-target">Gĩkũyũ</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%97%E0%AA%A4%E0%AA%BF%E0%AA%A8%E0%AA%BE_%E0%AA%A8%E0%AA%BF%E0%AA%AF%E0%AA%AE%E0%AB%8B" title="ગતિના નિયમો – Gujarati" lang="gu" hreflang="gu" data-title="ગતિના નિયમો" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%89%B4%ED%84%B4_%EC%9A%B4%EB%8F%99_%EB%B2%95%EC%B9%99" title="뉴턴 운동 법칙 – Korean" lang="ko" hreflang="ko" data-title="뉴턴 운동 법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%86%D5%B5%D5%B8%D6%82%D5%BF%D5%B8%D5%B6%D5%AB_%D6%85%D6%80%D5%A5%D5%B6%D6%84%D5%B6%D5%A5%D6%80" title="Նյուտոնի օրենքներ – Armenian" lang="hy" hreflang="hy" data-title="Նյուտոնի օրենքներ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A5%82%E0%A4%9F%E0%A4%A8_%E0%A4%95%E0%A5%87_%E0%A4%97%E0%A4%A4%E0%A4%BF_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="न्यूटन के गति नियम – Hindi" lang="hi" hreflang="hi" data-title="न्यूटन के गति नियम" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Newtonovi_zakoni_gibanja" title="Newtonovi zakoni gibanja – Croatian" lang="hr" hreflang="hr" data-title="Newtonovi zakoni gibanja" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Legi_di_Newton" title="Legi di Newton – Ido" lang="io" hreflang="io" data-title="Legi di Newton" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hukum_gerak_Newton" title="Hukum gerak Newton – Indonesian" lang="id" hreflang="id" data-title="Hukum gerak Newton" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Leges_de_Newton" title="Leges de Newton – Interlingua" lang="ia" hreflang="ia" data-title="Leges de Newton" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Imithetho_yomdiki_kaNewton" title="Imithetho yomdiki kaNewton – Zulu" lang="zu" hreflang="zu" data-title="Imithetho yomdiki kaNewton" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target">IsiZulu</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/L%C3%B6gm%C3%A1l_Newtons" title="Lögmál Newtons – Icelandic" lang="is" hreflang="is" data-title="Lögmál Newtons" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Principi_della_dinamica" title="Principi della dinamica – Italian" lang="it" hreflang="it" data-title="Principi della dinamica" 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%97%D7%95%D7%A7%D7%99_%D7%94%D7%AA%D7%A0%D7%95%D7%A2%D7%94_%D7%A9%D7%9C_%D7%A0%D7%99%D7%95%D7%98%D7%95%D7%9F" title="חוקי התנועה של ניוטון – Hebrew" lang="he" hreflang="he" data-title="חוקי התנועה של ניוטון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Newton_ciiduu_pa%C9%A3t%CA%8B_nd%C9%A9_nd%C9%A9" title="Newton ciiduu paɣtʋ ndɩ ndɩ – Kabiye" lang="kbp" hreflang="kbp" data-title="Newton ciiduu paɣtʋ ndɩ ndɩ" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target">Kabɩyɛ</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A8%E0%B3%8D%E0%B2%AF%E0%B3%82%E0%B2%9F%E0%B2%A8%E0%B3%8D%E2%80%8D%E0%B2%A8_%E0%B2%9A%E0%B2%B2%E0%B2%A8%E0%B3%86%E0%B2%AF_%E0%B2%A8%E0%B2%BF%E0%B2%AF%E0%B2%AE%E0%B2%97%E0%B2%B3%E0%B3%81" title="ನ್ಯೂಟನ್‍ನ ಚಲನೆಯ ನಿಯಮಗಳು – Kannada" lang="kn" hreflang="kn" data-title="ನ್ಯೂಟನ್‍ನ ಚಲನೆಯ ನಿಯಮಗಳು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9C%E1%83%98%E1%83%A3%E1%83%A2%E1%83%9D%E1%83%9C%E1%83%98%E1%83%A1_%E1%83%99%E1%83%90%E1%83%9C%E1%83%9D%E1%83%9C%E1%83%94%E1%83%91%E1%83%98" title="ნიუტონის კანონები – Georgian" lang="ka" hreflang="ka" data-title="ნიუტონის კანონები" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD_%D0%B7%D0%B0%D2%A3%D0%B4%D0%B0%D1%80%D1%8B" title="Ньютон заңдары – Kazakh" lang="kk" hreflang="kk" data-title="Ньютон заңдары" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Lwa_mouvman_Newton" title="Lwa mouvman Newton – Haitian Creole" lang="ht" hreflang="ht" data-title="Lwa mouvman Newton" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Lalwa_di_mouvman_di_Newton" title="Lalwa di mouvman di Newton – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Lalwa di mouvman di Newton" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Leges_motus_Newtoni" title="Leges motus Newtoni – Latin" lang="la" hreflang="la" data-title="Leges motus Newtoni" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C5%85%C5%ABtona_likumi" title="Ņūtona likumi – Latvian" lang="lv" hreflang="lv" data-title="Ņūtona likumi" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Niutono_d%C4%97sniai" title="Niutono dėsniai – Lithuanian" lang="lt" hreflang="lt" data-title="Niutono dėsniai" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/W%C3%A8tte_van_Newton" title="Wètte van Newton – Limburgish" lang="li" hreflang="li" data-title="Wètte van Newton" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Newton_t%C3%B6rv%C3%A9nyei" title="Newton törvényei – Hungarian" lang="hu" hreflang="hu" data-title="Newton törvényei" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mai mw-list-item"><a href="https://mai.wikipedia.org/wiki/%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%9F%E0%A4%A8%E0%A4%95_%E0%A4%97%E0%A4%A4%E0%A4%BF_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="न्युटनक गति नियम – Maithili" lang="mai" hreflang="mai" data-title="न्युटनक गति नियम" data-language-autonym="मैथिली" data-language-local-name="Maithili" class="interlanguage-link-target">मैथिली</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%8A%D1%83%D1%82%D0%BD%D0%BE%D0%B2%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8" title="Њутнови закони – Macedonian" lang="mk" hreflang="mk" data-title="Њутнови закони" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A8%E0%B5%8D%E0%B4%AF%E0%B5%82%E0%B4%9F%E0%B5%8D%E0%B4%9F%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%86_%E0%B4%9A%E0%B4%B2%E0%B4%A8%E0%B4%A8%E0%B4%BF%E0%B4%AF%E0%B4%AE%E0%B4%99%E0%B5%8D%E0%B4%99%E0%B5%BE" title="ന്യൂട്ടന്റെ ചലനനിയമങ്ങൾ – Malayalam" lang="ml" hreflang="ml" data-title="ന്യൂട്ടന്റെ ചലനനിയമങ്ങൾ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A5%82%E0%A4%9F%E0%A4%A8%E0%A4%9A%E0%A5%87_%E0%A4%97%E0%A4%A4%E0%A5%80%E0%A4%9A%E0%A5%87_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="न्यूटनचे गतीचे नियम – Marathi" lang="mr" hreflang="mr" data-title="न्यूटनचे गतीचे नियम" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Hukum-hukum_gerakan_Newton" title="Hukum-hukum gerakan Newton – Malay" lang="ms" hreflang="ms" data-title="Hukum-hukum gerakan Newton" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/Newton_%C3%B4ng-d%C3%B4ng_d%C3%AAng-l%C5%ADk" title="Newton ông-dông dêng-lŭk – Mindong" lang="cdo" hreflang="cdo" data-title="Newton ông-dông dêng-lŭk" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target">閩東語 / Mìng-dĕ̤ng-ngṳ̄</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D1%8B_%D1%85%D1%83%D1%83%D0%BB%D0%B8%D1%83%D0%B4" title="Ньютоны хуулиуд – Mongolian" lang="mn" hreflang="mn" data-title="Ньютоны хуулиуд" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%94%E1%80%9A%E1%80%B0%E1%80%90%E1%80%94%E1%80%BA%E1%81%8F_%E1%80%9B%E1%80%BD%E1%80%B1%E1%80%B7%E1%80%9C%E1%80%BB%E1%80%AC%E1%80%B8%E1%80%99%E1%80%BE%E1%80%AF%E1%80%86%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%9B%E1%80%AC_%E1%80%94%E1%80%AD%E1%80%9A%E1%80%AC%E1%80%99%E1%80%99%E1%80%BB%E1%80%AC%E1%80%B8" title="နယူတန်၏ ရွေ့လျားမှုဆိုင်ရာ နိယာမများ – Burmese" lang="my" hreflang="my" data-title="နယူတန်၏ ရွေ့လျားမှုဆိုင်ရာ နိယာမများ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wetten_van_Newton" title="Wetten van Newton – Dutch" lang="nl" hreflang="nl" data-title="Wetten van Newton" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nds-nl mw-list-item"><a href="https://nds-nl.wikipedia.org/wiki/Newton_zien_wetten" title="Newton zien wetten – Low Saxon" lang="nds-NL" hreflang="nds-NL" data-title="Newton zien wetten" data-language-autonym="Nedersaksies" data-language-local-name="Low Saxon" class="interlanguage-link-target">Nedersaksies</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%9F%E0%A4%A8%E0%A4%95%E0%A5%8B_%E0%A4%97%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A4%BE_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE%E0%A4%B9%E0%A4%B0%E0%A5%82" title="न्युटनको गतिका नियमहरू – Nepali" lang="ne" hreflang="ne" data-title="न्युटनको गतिका नियमहरू" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E5%8A%9B%E5%AD%A6" title="ニュートン力学 – Japanese" lang="ja" hreflang="ja" data-title="ニュートン力学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Newtons_bevegelseslover" title="Newtons bevegelseslover – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Newtons bevegelseslover" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Newtons_r%C3%B8rslelover" title="Newtons rørslelover – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Newtons rørslelover" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Leis_de_Newton" title="Leis de Newton – Occitan" lang="oc" hreflang="oc" data-title="Leis de Newton" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Newton_qonunlari" title="Newton qonunlari – Uzbek" lang="uz" hreflang="uz" data-title="Newton qonunlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A8%E0%A8%BF%E0%A8%8A%E0%A8%9F%E0%A8%A8_%E0%A8%A6%E0%A9%87_%E0%A8%97%E0%A8%A4%E0%A9%80_%E0%A8%A6%E0%A9%87_%E0%A8%A8%E0%A8%BF%E0%A8%AF%E0%A8%AE" title="ਨਿਊਟਨ ਦੇ ਗਤੀ ਦੇ ਨਿਯਮ – Punjabi" lang="pa" hreflang="pa" data-title="ਨਿਊਟਨ ਦੇ ਗਤੀ ਦੇ ਨਿਯਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%86%DB%8C%D9%88%D9%B9%D9%86_%D8%AF%DB%92_%DA%86%D9%84%D9%86_%D8%AF%DB%92_%D9%82%D9%86%D9%88%D9%86" title="نیوٹن دے چلن دے قنون – Western Punjabi" lang="pnb" hreflang="pnb" data-title="نیوٹن دے چلن دے قنون" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D8%AE%D9%88%DA%81%DA%9A%D8%AA_%D9%BE%D9%87_%D8%A7%DA%93%D9%87_%D8%AF_%D9%86%DB%8C%D9%88%D9%BC%D9%86_%D9%82%D9%88%D8%A7%D9%86%DB%8C%D9%86" title="د خوځښت په اړه د نیوټن قوانین – Pashto" lang="ps" hreflang="ps" data-title="د خوځښت په اړه د نیوټن قوانین" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Nyuutan_laa_a_muoshan" title="Nyuutan laa a muoshan – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Nyuutan laa a muoshan" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%85%E1%9F%92%E1%9E%94%E1%9E%B6%E1%9E%94%E1%9F%8B%E1%9E%85%E1%9E%9B%E1%9E%93%E1%9E%B6%E1%9E%9A%E1%9E%94%E1%9E%9F%E1%9F%8B%E1%9E%89%E1%9E%BC%E1%9E%8F%E1%9E%BB%E1%9E%93" title="ច្បាប់ចលនារបស់ញូតុន – Khmer" lang="km" hreflang="km" data-title="ច្បាប់ចលនារបស់ញូតុន" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Lej_d%C3%ABl_moviment_%C3%ABd_Newton" title="Lej dël moviment ëd Newton – Piedmontese" lang="pms" hreflang="pms" data-title="Lej dël moviment ëd Newton" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zasady_dynamiki_Newtona" title="Zasady dynamiki Newtona – Polish" lang="pl" hreflang="pl" data-title="Zasady dynamiki Newtona" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Leis_de_Newton" title="Leis de Newton – Portuguese" lang="pt" hreflang="pt" data-title="Leis de Newton" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Legile_lui_Newton" title="Legile lui Newton – Romanian" lang="ro" hreflang="ro" data-title="Legile lui Newton" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D0%BE%D0%B2%D1%8B_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD%D1%8B_%D1%80%D1%83%D1%88%D0%B0%D0%BD%D1%8F" title="Ньютоновы законы рушаня – Rusyn" lang="rue" hreflang="rue" data-title="Ньютоновы законы рушаня" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target">Русиньскый</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD%D1%8B_%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D0%B0" title="Законы Ньютона – Russian" lang="ru" hreflang="ru" data-title="Законы Ньютона" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD_%D1%81%D0%BE%D0%BA%D1%83%D0%BE%D0%BD%D0%BD%D0%B0%D1%80%D0%B0" title="Ньютон сокуоннара – Yakut" lang="sah" hreflang="sah" data-title="Ньютон сокуоннара" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-skr mw-list-item"><a href="https://skr.wikipedia.org/wiki/%D9%86%DB%8C%D9%88%D9%B9%D9%86_%D8%AF%D8%A7_%D9%BE%DB%81%D9%84%D8%A7_%D9%82%D9%86%D9%88%D9%86" title="نیوٹن دا پہلا قنون – Saraiki" lang="skr" hreflang="skr" data-title="نیوٹن دا پہلا قنون" data-language-autonym="سرائیکی" data-language-local-name="Saraiki" class="interlanguage-link-target">سرائیکی</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Ligjet_e_Njutonit" title="Ligjet e Njutonit – Albanian" lang="sq" hreflang="sq" data-title="Ligjet e Njutonit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Liggi_di_Newton" title="Liggi di Newton – Sicilian" lang="scn" hreflang="scn" data-title="Liggi di Newton" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%A0%E0%B6%BD%E0%B7%92%E0%B6%AD%E0%B6%BA_%E0%B6%B4%E0%B7%92%E0%B7%85%E0%B7%92%E0%B6%B6%E0%B6%B3_%E0%B6%B1%E0%B7%92%E0%B7%80%E0%B7%8A%E0%B6%A7%E0%B6%B1%E0%B7%8A_%E0%B6%B1%E0%B7%92%E0%B6%BA%E0%B6%B8" title="චලිතය පිළිබඳ නිව්ටන් නියම – Sinhala" lang="si" hreflang="si" data-title="චලිතය පිළිබඳ නිව්ටන් නියම" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion – Simple English" lang="en-simple" hreflang="en-simple" data-title="Newton's laws of motion" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D9%86%D9%8A%D9%88%D9%BD%D9%86_%D8%AC%D8%A7_%D8%AD%D8%B1%DA%AA%D8%AA_%D8%AC%D8%A7_%D9%82%D8%A7%D9%86%D9%88%D9%86" title="نيوٽن جا حرڪت جا قانون – Sindhi" lang="sd" hreflang="sd" data-title="نيوٽن جا حرڪت جا قانون" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target">سنڌي</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Newtonove_pohybov%C3%A9_z%C3%A1kony" title="Newtonove pohybové zákony – Slovak" lang="sk" hreflang="sk" data-title="Newtonove pohybové zákony" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Newtonovi_zakoni_gibanja" title="Newtonovi zakoni gibanja – Slovenian" lang="sl" hreflang="sl" data-title="Newtonovi zakoni gibanja" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DB%8C%D8%A7%D8%B3%D8%A7%DA%A9%D8%A7%D9%86%DB%8C_%D8%AC%D9%88%D9%88%DA%B5%DB%95%DB%8C_%D9%86%DB%8C%D9%88%D8%AA%D9%86" title="یاساکانی جووڵەی نیوتن – Central Kurdish" lang="ckb" hreflang="ckb" data-title="یاساکانی جووڵەی نیوتن" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%8A%D1%83%D1%82%D0%BD%D0%BE%D0%B2%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8" title="Њутнови закони – Serbian" lang="sr" hreflang="sr" data-title="Њутнови закони" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Newtonovi_zakoni_kretanja" title="Newtonovi zakoni kretanja – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Newtonovi zakoni kretanja" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Hukum_gerak_Newton" title="Hukum gerak Newton – Sundanese" lang="su" hreflang="su" data-title="Hukum gerak Newton" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a 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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"><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 mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"> <ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a class="mw-selflink selflink">Newton's laws of motion</a></div></li> <li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a> <div class="plainlist"><ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="/wiki/Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="/wiki/Appell%27s_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Core topics</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Damping" title="Damping">Damping</a></li> <li><a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li> <li><a href="/wiki/Equations_of_motion" title="Equations of motion">Equations of motion</a></li> <li><a href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion">Euler's laws of motion</a></li> <li><a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a></li> <li><a href="/wiki/Friction" title="Friction">Friction</a></li> <li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul> </div> <ul><li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a> / <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Motion" title="Motion">Motion</a> (<a href="/wiki/Linear_motion" title="Linear motion">linear</a>)</li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a></li> <li><a class="mw-selflink selflink">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 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.navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classical_mechanics" title="Template:Classical mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classical_mechanics" title="Template talk:Classical mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics" title="Special:EditPage/Template:Classical mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> Newton's laws of motion are three <a href="/wiki/Physical_laws" class="mw-redirect" title="Physical laws">physical laws</a> that describe the relationship between the <a href="/wiki/Motion" title="Motion">motion</a> of an object and the <a href="/wiki/Force" title="Force">forces</a> acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: <ol><li>A body remains at rest, or in motion at a constant speed in a straight line, except insofar as it is acted upon by a force.</li> <li>At any instant of time, the net force on a body is equal to the body's <a href="/wiki/Acceleration" title="Acceleration">acceleration</a> multiplied by its mass or, equivalently, the rate at which the body's <a href="/wiki/Momentum" title="Momentum">momentum</a> is changing with time.</li> <li>If two bodies exert forces on each other, these forces have the same magnitude but opposite directions.<a href="#cite_note-Thornton-1">[1]</a><a href="#cite_note-Cohen&Whitman-2">[2]</a></li></ol> The three laws of motion were first stated by <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> in his <a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Philosophiæ Naturalis Principia Mathematica</a> (Mathematical Principles of Natural Philosophy), originally published in 1687.<a href="#cite_note-3">[3]</a> Newton used them to investigate and explain the motion of many physical objects and systems. In the time since Newton, new insights, especially around the concept of energy, built the field of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> on his foundations. Limitations to Newton's laws have also been discovered; new theories are necessary when objects move at very high speeds (<a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>), are very massive (<a href="/wiki/General_relativity" title="General relativity">general relativity</a>), or are very small (<a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>). <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Prerequisites">Prerequisites</h2></div> Newton's laws are often stated in terms of point or particle masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each. For instance, the Earth and the Sun can both be approximated as pointlike when considering the orbit of the former around the latter, but the Earth is not pointlike when considering activities on its surface.<a href="#cite_note-6">[note 1]</a> The mathematical description of motion, or <a href="/wiki/Kinematics" title="Kinematics">kinematics</a>, is based on the idea of specifying positions using numerical coordinates. Movement is represented by these numbers changing over time: a body's trajectory is represented by a function that assigns to each value of a time variable the values of all the position coordinates. The simplest case is one-dimensional, that is, when a body is constrained to move only along a straight line. Its position can then be given by a single number, indicating where it is relative to some chosen reference point. For example, a body might be free to slide along a track that runs left to right, and so its location can be specified by its distance from a convenient zero point, or <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a>, with negative numbers indicating positions to the left and positive numbers indicating positions to the right. If the body's location as a function of time is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c484de351ba40ccb9a5ad522c29c1aac5686c0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}">, then its average velocity over the time interval from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0768c0bd659f2f84fb5ef9f4b74f336123d915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}"> is<a href="#cite_note-Hughes-Hallett-7">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta s}{\Delta t}}={\frac {s(t_{1})-s(t_{0})}{t_{1}-t_{0}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>s</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>s</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta s}{\Delta t}}={\frac {s(t_{1})-s(t_{0})}{t_{1}-t_{0}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd4169dfba9f6e2b80a44a4810c33910e95150d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.872ex; height:6.009ex;" alt="{\displaystyle {\frac {\Delta s}{\Delta t}}={\frac {s(t_{1})-s(t_{0})}{t_{1}-t_{0}}}.}">Here, the Greek letter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"> (<a href="/wiki/Delta_(letter)" title="Delta (letter)">delta</a>) is used, per tradition, to mean "change in". A positive average velocity means that the position coordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> increases over the interval in question, a negative average velocity indicates a net decrease over that interval, and an average velocity of zero means that the body ends the time interval in the same place as it began. <a href="/wiki/Calculus" title="Calculus">Calculus</a> gives the means to define an instantaneous velocity, a measure of a body's speed and direction of movement at a single moment of time, rather than over an interval. One notation for the instantaneous velocity is to replace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"> with the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}">, for example,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\frac {ds}{dt}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>s</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {ds}{dt}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f22a6ada5b3f3d8a9450c9b85fd2d85adf0be76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.015ex; height:5.509ex;" alt="{\displaystyle v={\frac {ds}{dt}}.}">This denotes that the instantaneous velocity is the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the position with respect to time. It can roughly be thought of as the ratio between an infinitesimally small change in position <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f0fb36e4308227d3e4a1f809c2571ec02527100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.306ex; height:2.176ex;" alt="{\displaystyle ds}"> to the infinitesimally small time interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebee76a835701fd1f26047a09855f2ea36bb08fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.055ex; height:2.176ex;" alt="{\displaystyle dt}"> over which it occurs.<a href="#cite_note-Thompson-8">[7]</a> More carefully, the velocity and all other derivatives can be defined using the concept of a <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a>.<a href="#cite_note-Hughes-Hallett-7">[6]</a> A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"> has a limit of <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}"> at a given input value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}"> if the difference between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <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}"> can be made arbitrarily small by choosing an input sufficiently close to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}">. One writes, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{t\to t_{0}}f(t)=L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{t\to t_{0}}f(t)=L.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/643dacc66c1939696a39541ff8eb5219e3ab0103" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.305ex; height:4.343ex;" alt="{\displaystyle \lim _{t\to t_{0}}f(t)=L.}">Instantaneous velocity can be defined as the limit of the average velocity as the time interval shrinks to zero:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {ds}{dt}}=\lim _{\Delta t\to 0}{\frac {s(t+\Delta t)-s(t)}{\Delta t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>s</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {ds}{dt}}=\lim _{\Delta t\to 0}{\frac {s(t+\Delta t)-s(t)}{\Delta t}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31fa3ca5d0c15e8454f6eed22ae4a62f8311d222" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.474ex; height:6.009ex;" alt="{\displaystyle {\frac {ds}{dt}}=\lim _{\Delta t\to 0}{\frac {s(t+\Delta t)-s(t)}{\Delta t}}.}"> Acceleration is to velocity as velocity is to position: it is the derivative of the velocity with respect to time.<a href="#cite_note-10">[note 2]</a> Acceleration can likewise be defined as a limit:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {dv}{dt}}=\lim _{\Delta t\to 0}{\frac {v(t+\Delta t)-v(t)}{\Delta t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {dv}{dt}}=\lim _{\Delta t\to 0}{\frac {v(t+\Delta t)-v(t)}{\Delta t}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca268bf87ff7da2f042785bb0b2cb40a9b983237" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.913ex; height:6.009ex;" alt="{\displaystyle a={\frac {dv}{dt}}=\lim _{\Delta t\to 0}{\frac {v(t+\Delta t)-v(t)}{\Delta t}}.}">Consequently, the acceleration is the second derivative of position,<a href="#cite_note-Thompson-8">[7]</a> often written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}s}{dt^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>s</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}s}{dt^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47c32d30bc8ea8a898d72cb33b68c05a5308cc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:4.199ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}s}{dt^{2}}}}">. Position, when thought of as a displacement from an origin point, is a <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a>: a quantity with both magnitude and direction.<a href="#cite_note-:1-11">[9]</a>: 1  Velocity and acceleration are vector quantities as well. The mathematical tools of vector algebra provide the means to describe motion in two, three or more dimensions. Vectors are often denoted with an arrow, as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {s}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57ce74ecbbc0e198d0bb4b1b2ffc0cb9669b72e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {s}}}">, or in bold typeface, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {s}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e910e1ee0ff1b7d304b2a02dd802d23aac1e1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle {\bf {s}}}">. Often, vectors are represented visually as arrows, with the direction of the vector being the direction of the arrow, and the magnitude of the vector indicated by the length of the arrow. Numerically, a vector can be represented as a list; for example, a body's velocity vector might be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =(\mathrm {3~m/s} ,\mathrm {4~m/s} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mtext> </mtext> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">s</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mtext> </mtext> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =(\mathrm {3~m/s} ,\mathrm {4~m/s} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6522522fcb88bd97f1d71fa8ef069d82e3c7ad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.868ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} =(\mathrm {3~m/s} ,\mathrm {4~m/s} )}">, indicating that it is moving at 3 metres per second along the horizontal axis and 4 metres per second along the vertical axis. The same motion described in a different <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> will be represented by different numbers, and vector algebra can be used to translate between these alternatives.<a href="#cite_note-:1-11">[9]</a>: 4  The study of mechanics is complicated by the fact that household words like energy are used with a technical meaning.<a href="#cite_note-12">[10]</a> Moreover, words which are synonymous in everyday speech are not so in physics: force is not the same as power or pressure, for example, and mass has a different meaning than weight.<a href="#cite_note-13">[11]</a><a href="#cite_note-openstax-14">[12]</a>: 150  The physics concept of force makes quantitative the everyday idea of a push or a pull. Forces in Newtonian mechanics are often due to strings and ropes, friction, muscle effort, gravity, and so forth. Like displacement, velocity, and acceleration, force is a vector quantity. <div class="mw-heading mw-heading2"><h2 id="Laws">Laws</h2></div> <div class="mw-heading mw-heading3"><h3 id="First_law">First law</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Skylab_and_Earth_Limb_-_GPN-2000-001055.jpg" class="mw-file-description"><img alt="see caption" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Skylab_and_Earth_Limb_-_GPN-2000-001055.jpg/220px-Skylab_and_Earth_Limb_-_GPN-2000-001055.jpg" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Skylab_and_Earth_Limb_-_GPN-2000-001055.jpg/330px-Skylab_and_Earth_Limb_-_GPN-2000-001055.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Skylab_and_Earth_Limb_-_GPN-2000-001055.jpg/440px-Skylab_and_Earth_Limb_-_GPN-2000-001055.jpg 2x" data-file-width="3856" data-file-height="3870" /></a><figcaption>Artificial satellites move along curved <a href="/wiki/Orbit" title="Orbit">orbits</a>, rather than in straight lines, because of the Earth's <a href="/wiki/Gravity" title="Gravity">gravity</a>.</figcaption></figure> Translated from Latin, Newton's first law reads, <dl><dd>Every object perseveres in its state of rest, or of uniform motion in a right line, except insofar as it is compelled to change that state by forces impressed thereon.<a href="#cite_note-19">[note 3]</a></dd></dl> Newton's first law expresses the principle of <a href="/wiki/Inertia" title="Inertia">inertia</a>: the natural behavior of a body is to move in a straight line at constant speed. A body's motion preserves the status quo, but external forces can perturb this. The modern understanding of Newton's first law is that no <a href="/wiki/Inertial_observer" class="mw-redirect" title="Inertial observer">inertial observer</a> is privileged over any other. The concept of an inertial observer makes quantitative the everyday idea of feeling no effects of motion. For example, a person standing on the ground watching a train go past is an inertial observer. If the observer on the ground sees the train moving smoothly in a straight line at a constant speed, then a passenger sitting on the train will also be an inertial observer: the train passenger feels no motion. The principle expressed by Newton's first law is that there is no way to say which inertial observer is "really" moving and which is "really" standing still. One observer's state of rest is another observer's state of uniform motion in a straight line, and no experiment can deem either point of view to be correct or incorrect. There is no absolute standard of rest.<a href="#cite_note-20">[17]</a><a href="#cite_note-:0-16">[14]</a>: 62–63 <a href="#cite_note-:2-21">[18]</a>: 7–9  Newton himself believed that <a href="/wiki/Absolute_space_and_time" title="Absolute space and time">absolute space and time</a> existed, but that the only measures of space or time accessible to experiment are relative.<a href="#cite_note-22">[19]</a> <div class="mw-heading mw-heading3"><h3 id="Second_law">Second law</h3></div> <dl><dd>The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.<a href="#cite_note-:0-16">[14]</a>: 114 </dd></dl> By "motion", Newton meant the quantity now called <a href="/wiki/Momentum" title="Momentum">momentum</a>, which depends upon the amount of matter contained in a body, the speed at which that body is moving, and the direction in which it is moving.<a href="#cite_note-23">[20]</a> In modern notation, the momentum of a body is the product of its mass and its velocity: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =m\mathbf {v} \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =m\mathbf {v} \,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4c38d9148891f419dc63b8de193b40e01cbda2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.069ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} =m\mathbf {v} \,,}"> where all three quantities can change over time. Newton's second law, in modern form, states that the time derivative of the momentum is the force: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}\,.}"> <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"> <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> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99ecbac6df650ec669de4173a064ab3824920644" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.352ex; height:5.509ex;" alt="{\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}\,.}"> If the mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> does not change with time, then the derivative acts only upon the velocity, and so the force equals the product of the mass and the time derivative of the velocity, which is the acceleration:<a href="#cite_note-24">[21]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =m{\frac {d\mathbf {v} }{dt}}=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"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m{\frac {d\mathbf {v} }{dt}}=m\mathbf {a} \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45cf8f928f00376240c455bf40fa783b8980f351" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.757ex; height:5.509ex;" alt="{\displaystyle \mathbf {F} =m{\frac {d\mathbf {v} }{dt}}=m\mathbf {a} \,.}"> As the acceleration is the second derivative of position with respect to time, this can also be written <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =m{\frac {d^{2}\mathbf {s} }{dt^{2}}}.}"> <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"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</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 \mathbf {F} =m{\frac {d^{2}\mathbf {s} }{dt^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7204eb97b64aea4139eb902efa1f6dacd91a1a19" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.632ex; height:6.009ex;" alt="{\displaystyle \mathbf {F} =m{\frac {d^{2}\mathbf {s} }{dt^{2}}}.}"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Free_body1.3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Free_body1.3.svg/220px-Free_body1.3.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Free_body1.3.svg/330px-Free_body1.3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Free_body1.3.svg/440px-Free_body1.3.svg.png 2x" data-file-width="512" data-file-height="342" /></a><figcaption>A <a href="/wiki/Free_body_diagram" title="Free body diagram">free body diagram</a> for a block on an inclined plane, illustrating the <a href="/wiki/Normal_force" title="Normal force">normal force</a> perpendicular to the plane (N), the downward force of gravity (mg), and a force f along the direction of the plane that could be applied, for example, by friction or a string</figcaption></figure> The forces acting on a body <a href="/wiki/Euclidean_vector#Addition_and_subtraction" title="Euclidean vector">add as vectors</a>, and so the total force on a body depends upon both the magnitudes and the directions of the individual forces. When the net force on a body is equal to zero, then by Newton's second law, the body does not accelerate, and it is said to be in <a href="/wiki/Mechanical_equilibrium" title="Mechanical equilibrium">mechanical equilibrium</a>. A state of mechanical equilibrium is stable if, when the position of the body is changed slightly, the body remains near that equilibrium. Otherwise, the equilibrium is unstable. A common visual representation of forces acting in concert is the <a href="/wiki/Free_body_diagram" title="Free body diagram">free body diagram</a>, which schematically portrays a body of interest and the forces applied to it by outside influences.<a href="#cite_note-25">[22]</a> For example, a free body diagram of a block sitting upon an <a href="/wiki/Inclined_plane" title="Inclined plane">inclined plane</a> can illustrate the combination of gravitational force, <a href="/wiki/Normal_force" title="Normal force">"normal" force</a>, friction, and string tension.<a href="#cite_note-27">[note 4]</a> Newton's second law is sometimes presented as a definition of force, i.e., a force is that which exists when an inertial observer sees a body accelerating. In order for this to be more than a <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautology</a> — acceleration implies force, force implies acceleration — some other statement about force must also be made. For example, an equation detailing the force might be specified, like <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a>. By inserting such an expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da18bef8c979f3548bb0d8976f5844012d7b8256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }"> into Newton's second law, an equation with predictive power can be written.<a href="#cite_note-31">[note 5]</a> Newton's second law has also been regarded as setting out a research program for physics, establishing that important goals of the subject are to identify the forces present in nature and to catalogue the constituents of matter.<a href="#cite_note-:0-16">[14]</a>: 134 <a href="#cite_note-FLS-29">[25]</a>: 12-2  <div class="mw-heading mw-heading3"><h3 id="Third_law">Third law</h3></div> <dl><dd>To every action, there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.<a href="#cite_note-:0-16">[14]</a>: 116 </dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Iridium-1_Launch_(32312419215).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Iridium-1_Launch_%2832312419215%29.jpg/170px-Iridium-1_Launch_%2832312419215%29.jpg" decoding="async" width="170" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Iridium-1_Launch_%2832312419215%29.jpg/255px-Iridium-1_Launch_%2832312419215%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Iridium-1_Launch_%2832312419215%29.jpg/340px-Iridium-1_Launch_%2832312419215%29.jpg 2x" data-file-width="2000" data-file-height="3000" /></a><figcaption><a href="/wiki/Rocket" title="Rocket">Rockets</a> work by producing a strong <a href="/wiki/Reaction_(physics)" title="Reaction (physics)">reaction force</a> downwards using <a href="/wiki/Rocket_engine" title="Rocket engine">rocket engines</a>. This pushes the rocket upwards, without regard to the ground or the <a href="/wiki/Atmosphere" title="Atmosphere">atmosphere</a>.</figcaption></figure> Overly brief paraphrases of the third law, like "action equals <a href="/wiki/Reaction_(physics)" title="Reaction (physics)">reaction</a>" might have caused confusion among generations of students: the "action" and "reaction" apply to different bodies. For example, consider a book at rest on a table. The Earth's gravity pulls down upon the book. The "reaction" to that "action" is not the support force from the table holding up the book, but the gravitational pull of the book acting on the Earth.<a href="#cite_note-38">[note 6]</a> Newton's third law relates to a more fundamental principle, the <a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">conservation of momentum</a>. The latter remains true even in cases where Newton's statement does not, for instance when <a href="/wiki/Force_field_(physics)" title="Force field (physics)">force fields</a> as well as material bodies carry momentum, and when momentum is defined properly, in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> as well.<a href="#cite_note-39">[note 7]</a> In Newtonian mechanics, if two bodies have momenta <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/279e30b6f8eba8d6ff75f984f9393eebc2ea6d6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.54ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} _{1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0503cf42886fcd7909a39b737ba760b0249b3ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.54ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} _{2}}"> respectively, then the total momentum of the pair is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =\mathbf {p} _{1}+\mathbf {p} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =\mathbf {p} _{1}+\mathbf {p} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a232997be44e469d18176f7c6469979cf03bbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.503ex; height:2.509ex;" alt="{\displaystyle \mathbf {p} =\mathbf {p} _{1}+\mathbf {p} _{2}}">, and the rate of change of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\mathbf {p} }{dt}}={\frac {d\mathbf {p} _{1}}{dt}}+{\frac {d\mathbf {p} _{2}}{dt}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {p} }{dt}}={\frac {d\mathbf {p} _{1}}{dt}}+{\frac {d\mathbf {p} _{2}}{dt}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bbd1f10c07f4a12de996016f000a4c1edab8ab0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.306ex; height:5.676ex;" alt="{\displaystyle {\frac {d\mathbf {p} }{dt}}={\frac {d\mathbf {p} _{1}}{dt}}+{\frac {d\mathbf {p} _{2}}{dt}}.}"> By Newton's second law, the first term is the total force upon the first body, and the second term is the total force upon the second body. If the two bodies are isolated from outside influences, the only force upon the first body can be that from the second, and vice versa. By Newton's third law, these forces have equal magnitude but opposite direction, so they cancel when added, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"> is constant. Alternatively, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"> is known to be constant, it follows that the forces have equal magnitude and opposite direction. <div class="mw-heading mw-heading3"><h3 id="Candidates_for_additional_laws">Candidates for additional laws</h3></div> Various sources have proposed elevating other ideas used in classical mechanics to the status of Newton's laws. For example, in Newtonian mechanics, the total mass of a body made by bringing together two smaller bodies is the sum of their individual masses. <a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Frank Wilczek</a> has suggested calling attention to this assumption by designating it "Newton's Zeroth Law".<a href="#cite_note-40">[33]</a> Another candidate for a "zeroth law" is the fact that at any instant, a body reacts to the forces applied to it at that instant.<a href="#cite_note-41">[34]</a> Likewise, the idea that forces add like vectors (or in other words obey the <a href="/wiki/Superposition_principle" title="Superposition principle">superposition principle</a>), and the idea that forces change the energy of a body, have both been described as a "fourth law".<a href="#cite_note-46">[note 8]</a> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div> The study of the behavior of massive bodies using Newton's laws is known as Newtonian mechanics. Some example problems in Newtonian mechanics are particularly noteworthy for conceptual or historical reasons. <div class="mw-heading mw-heading4"><h4 id="Uniformly_accelerated_motion">Uniformly accelerated motion</h4></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/Free_fall" title="Free fall">Free fall</a> and <a href="/wiki/Projectile_motion" title="Projectile motion">Projectile motion</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bouncing_ball_strobe_edit.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Bouncing_ball_strobe_edit.jpg/290px-Bouncing_ball_strobe_edit.jpg" decoding="async" width="290" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Bouncing_ball_strobe_edit.jpg/435px-Bouncing_ball_strobe_edit.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Bouncing_ball_strobe_edit.jpg/580px-Bouncing_ball_strobe_edit.jpg 2x" data-file-width="1800" data-file-height="1159" /></a><figcaption>A <a href="/wiki/Bouncing_ball" title="Bouncing ball">bouncing ball</a> photographed at 25 frames per second using a <a href="/wiki/Stroboscope" title="Stroboscope">stroboscopic flash</a>. In between bounces, the ball's height as a function of time is close to being a <a href="/wiki/Parabola" title="Parabola">parabola</a>, deviating from a parabolic arc because of air resistance, spin, and deformation into a non-spherical shape upon impact.</figcaption></figure> If a body falls from rest near the surface of the Earth, then in the absence of air resistance, it will accelerate at a constant rate. This is known as <a href="/wiki/Free_fall" title="Free fall">free fall</a>. The speed attained during free fall is proportional to the elapsed time, and the distance traveled is proportional to the square of the elapsed time.<a href="#cite_note-47">[39]</a> Importantly, the acceleration is the same for all bodies, independently of their mass. This follows from combining Newton's second law of motion with his <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">law of universal gravitation</a>. The latter states that the magnitude of the gravitational force from the Earth upon the body is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F={\frac {GMm}{r^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F={\frac {GMm}{r^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa486837554264344fb8f4dff3106f1a0715054f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.632ex; height:5.676ex;" alt="{\displaystyle F={\frac {GMm}{r^{2}}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> is the mass of the falling body, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is the mass of the Earth, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> is Newton's constant, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> is the distance from the center of the Earth to the body's location, which is very nearly the radius of the Earth. Setting this equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ma}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ma}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f38bd3876d01199d10b88b0b5c2eb64bbc86b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.27ex; height:1.676ex;" alt="{\displaystyle ma}">, the body's mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> cancels from both sides of the equation, leaving an acceleration that depends upon <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> can be taken to be constant. This particular value of acceleration is typically denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g={\frac {GM}{r^{2}}}\approx \mathrm {9.8~m/s^{2}} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>9.8</mn> <mtext> </mtext> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g={\frac {GM}{r^{2}}}\approx \mathrm {9.8~m/s^{2}} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f81f4af29f49765f29efaebacbc32b896b542485" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.686ex; height:5.676ex;" alt="{\displaystyle g={\frac {GM}{r^{2}}}\approx \mathrm {9.8~m/s^{2}} .}"> If the body is not released from rest but instead launched upwards and/or horizontally with nonzero velocity, then free fall becomes <a href="/wiki/Projectile_motion" title="Projectile motion">projectile motion</a>.<a href="#cite_note-48">[40]</a> When air resistance can be neglected, projectiles follow <a href="/wiki/Parabola" title="Parabola">parabola</a>-shaped trajectories, because gravity affects the body's vertical motion and not its horizontal. At the peak of the projectile's trajectory, its vertical velocity is zero, but its acceleration is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> downwards, as it is at all times. Setting the wrong vector equal to zero is a common confusion among physics students.<a href="#cite_note-49">[41]</a> <div class="mw-heading mw-heading4"><h4 id="Uniform_circular_motion">Uniform circular motion</h4></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/Circular_motion" title="Circular motion">Circular motion</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Binary_system_orbit_q%3D3_e%3D0.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/ae/Binary_system_orbit_q%3D3_e%3D0.gif" decoding="async" width="220" height="220" class="mw-file-element" data-file-width="220" data-file-height="220" /></a><figcaption>Two objects in uniform circular motion, orbiting around the <a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">barycenter</a> (center of mass of both objects)</figcaption></figure> When a body is in uniform circular motion, the force on it changes the direction of its motion but not its speed. For a body moving in a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> at a constant speed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}">, its acceleration has a magnitude<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {v^{2}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {v^{2}}{r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0220b1356204c98557a8b60d275196be3e9cf0c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.346ex; height:5.676ex;" alt="{\displaystyle a={\frac {v^{2}}{r}}}">and is directed toward the center of the circle.<a href="#cite_note-51">[note 9]</a> The force required to sustain this acceleration, called the <a href="/wiki/Centripetal_force" title="Centripetal force">centripetal force</a>, is therefore also directed toward the center of the circle and has magnitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mv^{2}/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mv^{2}/r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/778e7247959c5af93ea3197b7cf7569826979e12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.433ex; height:3.176ex;" alt="{\displaystyle mv^{2}/r}">. Many <a href="/wiki/Orbit" title="Orbit">orbits</a>, such as that of the Moon around the Earth, can be approximated by uniform circular motion. In such cases, the centripetal force is gravity, and by Newton's law of universal gravitation has magnitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GMm/r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>M</mi> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GMm/r^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f4258974f3b3e076be88d9948712ef36b986cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.575ex; height:3.176ex;" alt="{\displaystyle GMm/r^{2}}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is the mass of the larger body being orbited. Therefore, the mass of a body can be calculated from observations of another body orbiting around it.<a href="#cite_note-52">[43]</a>: 130  <a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a> is a <a href="/wiki/Thought_experiment" title="Thought experiment">thought experiment</a> that interpolates between projectile motion and uniform circular motion. A cannonball that is lobbed weakly off the edge of a tall cliff will hit the ground in the same amount of time as if it were dropped from rest, because the force of gravity only affects the cannonball's momentum in the downward direction, and its effect is not diminished by horizontal movement. If the cannonball is launched with a greater initial horizontal velocity, then it will travel farther before it hits the ground, but it will still hit the ground in the same amount of time. However, if the cannonball is launched with an even larger initial velocity, then the curvature of the Earth becomes significant: the ground itself will curve away from the falling cannonball. A very fast cannonball will fall away from the inertial straight-line trajectory at the same rate that the Earth curves away beneath it; in other words, it will be in orbit (imagining that it is not slowed by air resistance or obstacles).<a href="#cite_note-53">[44]</a> <div class="mw-heading mw-heading4"><h4 id="Harmonic_motion">Harmonic motion</h4></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/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Animated-mass-spring.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/25/Animated-mass-spring.gif" decoding="async" width="160" height="320" class="mw-file-element" data-file-width="160" data-file-height="320" /></a><figcaption>An undamped <a href="/wiki/Spring%E2%80%93mass_system" class="mw-redirect" title="Spring–mass system">spring–mass system</a> undergoes simple harmonic motion.</figcaption></figure> Consider a body of mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> able to move along 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}"> axis, and suppose an equilibrium point exists at the position <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}">. That is, at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}">, the net force upon the body is the zero vector, and by Newton's second law, the body will not accelerate. If the force upon the body is proportional to the displacement from the equilibrium point, and directed to the equilibrium point, then the body will perform <a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">simple harmonic motion</a>. Writing the force as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=-kx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=-kx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aace79ea36db4ebc9f83a00c4198f4c054f5b4fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.188ex; height:2.343ex;" alt="{\displaystyle F=-kx}">, Newton's second law becomes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-kx\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-kx\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90b9df5b7569e73835a019b647950437b46b52d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.96ex; height:6.009ex;" alt="{\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-kx\,.}"> This differential equation has the solution <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=A\cos \omega t+B\sin \omega t\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>B</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=A\cos \omega t+B\sin \omega t\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d211558e31c7d28cb619cd78ba81210c676023e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.898ex; height:2.843ex;" alt="{\displaystyle x(t)=A\cos \omega t+B\sin \omega t\,}"> where the frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {k/m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {k/m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16585c1e039075df0261172c0f8aa08fdc09dc68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.738ex; height:4.843ex;" alt="{\displaystyle {\sqrt {k/m}}}">, and the constants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> can be calculated knowing, for example, the position and velocity the body has at a given time, like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}">. One reason that the harmonic oscillator is a conceptually important example is that it is good approximation for many systems near a stable mechanical equilibrium.<a href="#cite_note-55">[note 10]</a> For example, a <a href="/wiki/Pendulum" title="Pendulum">pendulum</a> has a stable equilibrium in the vertical position: if motionless there, it will remain there, and if pushed slightly, it will swing back and forth. Neglecting air resistance and friction in the pivot, the force upon the pendulum is gravity, and Newton's second law becomes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}=-{\frac {g}{L}}\sin \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}=-{\frac {g}{L}}\sin \theta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea32e4dbf842c0151ec6ba5ed6011aa5d264afa5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.892ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}=-{\frac {g}{L}}\sin \theta ,}">where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> is the length of the pendulum and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> is its angle from the vertical. When 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 }"> is small, the <a href="/wiki/Sine_and_cosine" title="Sine and cosine">sine</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> is nearly equal to <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 }"> (see <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>), and so this expression simplifies to the equation for a simple harmonic oscillator with frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\sqrt {g/L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {g/L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a199298fdb932aaff137d6c4df7ac778c00159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.729ex; height:4.843ex;" alt="{\displaystyle \omega ={\sqrt {g/L}}}">. A harmonic oscillator can be damped, often by friction or viscous drag, in which case energy bleeds out of the oscillator and the amplitude of the oscillations decreases over time. Also, a harmonic oscillator can be driven by an applied force, which can lead to the phenomenon of <a href="/wiki/Resonance" title="Resonance">resonance</a>.<a href="#cite_note-56">[46]</a> <div class="mw-heading mw-heading4"><h4 id="Objects_with_variable_mass">Objects with variable mass</h4></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/Variable-mass_system" title="Variable-mass system">Variable-mass system</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Space_Shuttle_Atlantis_launches_from_KSC_on_STS-132_side_view.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Space_Shuttle_Atlantis_launches_from_KSC_on_STS-132_side_view.jpg/220px-Space_Shuttle_Atlantis_launches_from_KSC_on_STS-132_side_view.jpg" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Space_Shuttle_Atlantis_launches_from_KSC_on_STS-132_side_view.jpg/330px-Space_Shuttle_Atlantis_launches_from_KSC_on_STS-132_side_view.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Space_Shuttle_Atlantis_launches_from_KSC_on_STS-132_side_view.jpg/440px-Space_Shuttle_Atlantis_launches_from_KSC_on_STS-132_side_view.jpg 2x" data-file-width="3424" data-file-height="2739" /></a><figcaption>Rockets, like the <a href="/wiki/Space_Shuttle_Atlantis" title="Space Shuttle Atlantis">Space Shuttle Atlantis</a>, propel matter in one direction to push the craft in the other. This means that the mass being pushed, the rocket and its remaining onboard fuel supply, is constantly changing.</figcaption></figure> Newtonian physics treats matter as being neither created nor destroyed, though it may be rearranged. It can be the case that an object of interest gains or loses mass because matter is added to or removed from it. In such a situation, Newton's laws can be applied to the individual pieces of matter, keeping track of which pieces belong to the object of interest over time. For instance, if a rocket of mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec7adc70ea0e1d365e5f7244b6f7ddad9fbf485" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.091ex; height:2.843ex;" alt="{\displaystyle M(t)}">, moving at velocity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} (t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee594765ca30167f80394d3349307a445782d012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.06ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} (t)}">, ejects matter at a velocity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"> relative to the rocket, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =M{\frac {d\mathbf {v} }{dt}}-\mathbf {u} {\frac {dM}{dt}}\,}"> <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"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>M</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =M{\frac {d\mathbf {v} }{dt}}-\mathbf {u} {\frac {dM}{dt}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ce24484b2d0168f9b48f4278511138d9910153" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.893ex; height:5.509ex;" alt="{\displaystyle \mathbf {F} =M{\frac {d\mathbf {v} }{dt}}-\mathbf {u} {\frac {dM}{dt}}\,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da18bef8c979f3548bb0d8976f5844012d7b8256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }"> is the net external force (e.g., a planet's gravitational pull).<a href="#cite_note-Kleppner-26">[23]</a>: 139  <div class="mw-heading mw-heading2"><h2 id="Work_and_energy">Work and energy</h2></div> Physicists developed the concept of <a href="/wiki/Energy" title="Energy">energy</a> after Newton's time, but it has become an inseparable part of what is considered "Newtonian" physics. Energy can broadly be classified into <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic</a>, due to a body's motion, and <a href="/wiki/Potential_energy" title="Potential energy">potential</a>, due to a body's position relative to others. <a href="/wiki/Thermal_energy" title="Thermal energy">Thermal energy</a>, the energy carried by heat flow, is a type of kinetic energy not associated with the macroscopic motion of objects but instead with the movements of the atoms and molecules of which they are made. According to the <a href="/wiki/Work-energy_theorem" class="mw-redirect" title="Work-energy theorem">work-energy theorem</a>, when a force acts upon a body while that body moves along the line of the force, the force does work upon the body, and the amount of work done is equal to the change in the body's kinetic energy.<a href="#cite_note-59">[note 11]</a> In many cases of interest, the net work done by a force when a body moves in a closed loop — starting at a point, moving along some trajectory, and returning to the initial point — is zero. If this is the case, then the force can be written in terms of the <a href="/wiki/Gradient" title="Gradient">gradient</a> of a function called a <a href="/wiki/Scalar_potential" title="Scalar potential">scalar potential</a>:<a href="#cite_note-Boas-50">[42]</a>: 303  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =-\mathbf {\nabla } U\,.}"> <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="normal">∇</mi> </mrow> <mi>U</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =-\mathbf {\nabla } U\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/607d3633111a0851e55066c97e992cf95db3a8ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.342ex; height:2.343ex;" alt="{\displaystyle \mathbf {F} =-\mathbf {\nabla } U\,.}"> This is true for many forces including that of gravity, but not for friction; indeed, almost any problem in a mechanics textbook that does not involve friction can be expressed in this way.<a href="#cite_note-hand-finch-54">[45]</a>: 19  The fact that the force can be written in this way can be understood from the <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a>. Without friction to dissipate a body's energy into heat, the body's energy will trade between potential and (non-thermal) kinetic forms while the total amount remains constant. Any gain of kinetic energy, which occurs when the net force on the body accelerates it to a higher speed, must be accompanied by a loss of potential energy. So, the net force upon the body is determined by the manner in which the potential energy decreases. <div class="mw-heading mw-heading2"><h2 id="Rigid-body_motion_and_rotation">Rigid-body motion and rotation</h2></div> A rigid body is an object whose size is too large to neglect and which maintains the same shape over time. In Newtonian mechanics, the motion of a rigid body is often understood by separating it into movement of the body's <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> and movement around the center of mass. <div class="mw-heading mw-heading3"><h3 id="Center_of_mass">Center of mass</h3></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/Center_of_mass" title="Center of mass">Center of mass</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Masocentro1.jpg" class="mw-file-description"><img alt="Fork-cork-toothpick object balanced on a pen on the toothpick part" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Masocentro1.jpg/220px-Masocentro1.jpg" decoding="async" width="220" height="171" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Masocentro1.jpg/330px-Masocentro1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/Masocentro1.jpg/440px-Masocentro1.jpg 2x" data-file-width="1286" data-file-height="1000" /></a><figcaption>The total center of mass of the <a href="/wiki/Fork" title="Fork">forks</a>, <a href="/wiki/Stopper_(plug)" title="Stopper (plug)">cork</a>, and <a href="/wiki/Toothpick" title="Toothpick">toothpick</a> is on top of the pen's tip.</figcaption></figure> Significant aspects of the motion of an extended body can be understood by imagining the mass of that body concentrated to a single point, known as the center of mass. The location of a body's center of mass depends upon how that body's material is distributed. For a collection of pointlike objects with masses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1},\ldots ,m_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1},\ldots ,m_{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/044398703078442d1e457a82773416a8622a576f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.005ex; height:2.009ex;" alt="{\displaystyle m_{1},\ldots ,m_{N}}"> at positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{1},\ldots ,\mathbf {r} _{N}}"> <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"> <mn>1</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{1},\ldots ,\mathbf {r} _{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54c2b2de164ae3ddfff88535d11ba6be061f0161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.128ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{1},\ldots ,\mathbf {r} _{N}}">, the center of mass is located at <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} =\sum _{i=1}^{N}{\frac {m_{i}\mathbf {r} _{i}}{M}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </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>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mi>M</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} =\sum _{i=1}^{N}{\frac {m_{i}\mathbf {r} _{i}}{M}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9a51d32aa52dd373a603b8524ec1451b9ae98f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.069ex; height:7.343ex;" alt="{\displaystyle \mathbf {R} =\sum _{i=1}^{N}{\frac {m_{i}\mathbf {r} _{i}}{M}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is the total mass of the collection. In the absence of a net external force, the center of mass moves at a constant speed in a straight line. This applies, for example, to a collision between two bodies.<a href="#cite_note-60">[49]</a> If the total external force is not zero, then the center of mass changes velocity as though it were a point body of mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">. This follows from the fact that the internal forces within the collection, the forces that the objects exert upon each other, occur in balanced pairs by Newton's third law. In a system of two bodies with one much more massive than the other, the center of mass will approximately coincide with the location of the more massive body.<a href="#cite_note-:2-21">[18]</a>: 22–24  <div class="mw-heading mw-heading3"><h3 id="Rotational_analogues_of_Newton's_laws">Rotational analogues of Newton's laws</h3></div> When Newton's laws are applied to rotating extended bodies, they lead to new quantities that are analogous to those invoked in the original laws. The analogue of mass is the <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>, the counterpart of momentum is <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>, and the counterpart of force is <a href="/wiki/Torque" title="Torque">torque</a>. Angular momentum is calculated with respect to a reference point.<a href="#cite_note-61">[50]</a> If the displacement vector from a reference point to a body is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> and the body has momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }">, then the body's angular momentum with respect to that point is, using the vector <a href="/wiki/Cross_product" title="Cross product">cross product</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f896b5a7319a8643044a20bef2feab2ed55c406f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.781ex; height:2.509ex;" alt="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} .}"> Taking the time derivative of the angular momentum gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\mathbf {L} }{dt}}=\left({\frac {d\mathbf {r} }{dt}}\right)\times \mathbf {p} +\mathbf {r} \times {\frac {d\mathbf {p} }{dt}}=\mathbf {v} \times m\mathbf {v} +\mathbf {r} \times \mathbf {F} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {L} }{dt}}=\left({\frac {d\mathbf {r} }{dt}}\right)\times \mathbf {p} +\mathbf {r} \times {\frac {d\mathbf {p} }{dt}}=\mathbf {v} \times m\mathbf {v} +\mathbf {r} \times \mathbf {F} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb070677a561f320e7f520f413e8da5d67682123" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.894ex; height:6.176ex;" alt="{\displaystyle {\frac {d\mathbf {L} }{dt}}=\left({\frac {d\mathbf {r} }{dt}}\right)\times \mathbf {p} +\mathbf {r} \times {\frac {d\mathbf {p} }{dt}}=\mathbf {v} \times m\mathbf {v} +\mathbf {r} \times \mathbf {F} .}"> The first term vanishes because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c60398405d407b172aa51baf7d27eeb8687d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.451ex; height:1.676ex;" alt="{\displaystyle m\mathbf {v} }"> point in the same direction. The remaining term is the torque, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\tau } =\mathbf {r} \times \mathbf {F} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\tau } =\mathbf {r} \times \mathbf {F} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8f5495f78b4b87b0b45ed088817f1e44adf716" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.572ex; height:2.176ex;" alt="{\displaystyle \mathbf {\tau } =\mathbf {r} \times \mathbf {F} .}"> When the torque is zero, the angular momentum is constant, just as when the force is zero, the momentum is constant.<a href="#cite_note-:2-21">[18]</a>: 14–15  The torque can vanish even when the force is non-zero, if the body is located at the reference point (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea91abce79b986f8283514f440f89893d27b8a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.363ex; height:2.176ex;" alt="{\displaystyle \mathbf {r} =0}">) or if the force <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da18bef8c979f3548bb0d8976f5844012d7b8256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }"> and the displacement vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> are directed along the same line. The angular momentum of a collection of point masses, and thus of an extended body, is found by adding the contributions from each of the points. This provides a means to characterize a body's rotation about an axis, by adding up the angular momenta of its individual pieces. The result depends on the chosen axis, the shape of the body, and the rate of rotation.<a href="#cite_note-:2-21">[18]</a>: 28  <div class="mw-heading mw-heading3"><h3 id="Multi-body_gravitational_system">Multi-body gravitational system</h3></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/Three-body_problem" title="Three-body problem">Three-body problem</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Three-body_Problem_Animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Three-body_Problem_Animation.gif/220px-Three-body_Problem_Animation.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/fa/Three-body_Problem_Animation.gif 1.5x" data-file-width="300" data-file-height="300" /></a><figcaption>Animation of three points or bodies attracting to each other</figcaption></figure> Newton's law of universal gravitation states that any body attracts any other body along the straight line connecting them. The size of the attracting force is proportional to the product of their masses, and inversely proportional to the square of the distance between them. Finding the shape of the orbits that an inverse-square force law will produce is known as the <a href="/wiki/Kepler_problem" title="Kepler problem">Kepler problem</a>. The Kepler problem can be solved in multiple ways, including by demonstrating that the <a href="/wiki/Laplace%E2%80%93Runge%E2%80%93Lenz_vector" title="Laplace–Runge–Lenz vector">Laplace–Runge–Lenz vector</a> is constant,<a href="#cite_note-62">[51]</a> or by applying a duality transformation to a 2-dimensional harmonic oscillator.<a href="#cite_note-63">[52]</a> However it is solved, the result is that orbits will be <a href="/wiki/Conic_section" title="Conic section">conic sections</a>, that is, <a href="/wiki/Ellipse" title="Ellipse">ellipses</a> (including circles), <a href="/wiki/Parabola" title="Parabola">parabolas</a>, or <a href="/wiki/Hyperbola" title="Hyperbola">hyperbolas</a>. The <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">eccentricity</a> of the orbit, and thus the type of conic section, is determined by the energy and the angular momentum of the orbiting body. Planets do not have sufficient energy to escape the Sun, and so their orbits are ellipses, to a good approximation; because the planets pull on one another, actual orbits are not exactly conic sections. If a third mass is added, the Kepler problem becomes the three-body problem, which in general has no exact solution in <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed form</a>. That is, there is no way to start from the differential equations implied by Newton's laws and, after a finite sequence of standard mathematical operations, obtain equations that express the three bodies' motions over time.<a href="#cite_note-Barrow-Green1997-64">[53]</a><a href="#cite_note-Barrow-Green2008-65">[54]</a> <a href="/wiki/Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods</a> can be applied to obtain useful, albeit approximate, results for the three-body problem.<a href="#cite_note-66">[55]</a> The positions and velocities of the bodies can be stored in <a href="/wiki/Variable_(computer_science)" title="Variable (computer science)">variables</a> within a computer's memory; Newton's laws are used to calculate how the velocities will change over a short interval of time, and knowing the velocities, the changes of position over that time interval can be computed. This process is <a href="/wiki/Loop_(computing)" class="mw-redirect" title="Loop (computing)">looped</a> to calculate, approximately, the bodies' trajectories. Generally speaking, the shorter the time interval, the more accurate the approximation.<a href="#cite_note-67">[56]</a> <div class="mw-heading mw-heading2"><h2 id="Chaos_and_unpredictability">Chaos and unpredictability</h2></div> <div class="mw-heading mw-heading3"><h3 id="Nonlinear_dynamics">Nonlinear dynamics</h3></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/Chaos_theory" title="Chaos theory">Chaos theory</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Demonstrating_Chaos_with_a_Double_Pendulum.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Demonstrating_Chaos_with_a_Double_Pendulum.gif/220px-Demonstrating_Chaos_with_a_Double_Pendulum.gif" decoding="async" width="220" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Demonstrating_Chaos_with_a_Double_Pendulum.gif/330px-Demonstrating_Chaos_with_a_Double_Pendulum.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Demonstrating_Chaos_with_a_Double_Pendulum.gif/440px-Demonstrating_Chaos_with_a_Double_Pendulum.gif 2x" data-file-width="658" data-file-height="689" /></a><figcaption>Three double pendulums, initialized with almost exactly the same initial conditions, diverge over time.</figcaption></figure> Newton's laws of motion allow the possibility of <a href="/wiki/Chaos_theory" title="Chaos theory">chaos</a>.<a href="#cite_note-:3-68">[57]</a><a href="#cite_note-69">[58]</a> That is, qualitatively speaking, physical systems obeying Newton's laws can exhibit sensitive dependence upon their initial conditions: a slight change of the position or velocity of one part of a system can lead to the whole system behaving in a radically different way within a short time. Noteworthy examples include the three-body problem, the <a href="/wiki/Double_pendulum" title="Double pendulum">double pendulum</a>, <a href="/wiki/Dynamical_billiards" title="Dynamical billiards">dynamical billiards</a>, and the <a href="/wiki/Fermi%E2%80%93Pasta%E2%80%93Ulam%E2%80%93Tsingou_problem" title="Fermi–Pasta–Ulam–Tsingou problem">Fermi–Pasta–Ulam–Tsingou problem</a>. Newton's laws can be applied to <a href="/wiki/Fluid" title="Fluid">fluids</a> by considering a fluid as composed of infinitesimal pieces, each exerting forces upon neighboring pieces. The <a href="/wiki/Euler_equations_(fluid_dynamics)" title="Euler equations (fluid dynamics)">Euler momentum equation</a> is an expression of Newton's second law adapted to fluid dynamics.<a href="#cite_note-Zee2020-70">[59]</a><a href="#cite_note-71">[60]</a> A fluid is described by a velocity field, i.e., a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} (\mathbf {x} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} (\mathbf {x} ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b9c5ae6b8c41d39658de387807109d572b6dfb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.505ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} (\mathbf {x} ,t)}"> that assigns a velocity vector to each point in space and time. A small object being carried along by the fluid flow can change velocity for two reasons: first, because the velocity field at its position is changing over time, and second, because it moves to a new location where the velocity field has a different value. Consequently, when Newton's second law is applied to an infinitesimal portion of fluid, the acceleration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"> has two terms, a combination known as a <a href="/wiki/Material_derivative" title="Material derivative">total or material derivative</a>. The mass of an infinitesimal portion depends upon the fluid <a href="/wiki/Density" title="Density">density</a>, and there is a net force upon it if the fluid pressure varies from one side of it to another. Accordingly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =\mathbf {F} /m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\mathbf {F} /m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38034156225231c906b1c1a777fc77bbfde60f6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.284ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} =\mathbf {F} /m}"> becomes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial v}{\partial t}}+(\mathbf {\nabla } \cdot \mathbf {v} )\mathbf {v} =-{\frac {1}{\rho }}\mathbf {\nabla } P+\mathbf {f} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>P</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial v}{\partial t}}+(\mathbf {\nabla } \cdot \mathbf {v} )\mathbf {v} =-{\frac {1}{\rho }}\mathbf {\nabla } P+\mathbf {f} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d6eed08dc833e0f6e93333f8a48963eaa17ec9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.535ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial v}{\partial t}}+(\mathbf {\nabla } \cdot \mathbf {v} )\mathbf {v} =-{\frac {1}{\rho }}\mathbf {\nabla } P+\mathbf {f} ,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> is the density, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> is the pressure, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6194e680a4e7c521f2178c50eea302843a852d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }"> stands for an external influence like a gravitational pull. Incorporating the effect of <a href="/wiki/Viscosity" title="Viscosity">viscosity</a> turns the Euler equation into a <a href="/wiki/Navier%E2%80%93Stokes_equation" class="mw-redirect" title="Navier–Stokes equation">Navier–Stokes equation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial v}{\partial t}}+(\mathbf {\nabla } \cdot \mathbf {v} )\mathbf {v} =-{\frac {1}{\rho }}\mathbf {\nabla } P+\nu \nabla ^{2}\mathbf {v} +\mathbf {f} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>P</mi> <mo>+</mo> <mi>ν</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial v}{\partial t}}+(\mathbf {\nabla } \cdot \mathbf {v} )\mathbf {v} =-{\frac {1}{\rho }}\mathbf {\nabla } P+\nu \nabla ^{2}\mathbf {v} +\mathbf {f} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90597361faa5c92d5cd5df2af6d9f6df0226ade9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.009ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial v}{\partial t}}+(\mathbf {\nabla } \cdot \mathbf {v} )\mathbf {v} =-{\frac {1}{\rho }}\mathbf {\nabla } P+\nu \nabla ^{2}\mathbf {v} +\mathbf {f} ,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"> is the <a href="/wiki/Viscosity#Kinematic_viscosity" title="Viscosity">kinematic viscosity</a>.<a href="#cite_note-Zee2020-70">[59]</a> <div class="mw-heading mw-heading3"><h3 id="Singularities">Singularities</h3></div> It is mathematically possible for a collection of point masses, moving in accord with Newton's laws, to launch some of themselves away so forcefully that they fly off to infinity in a finite time.<a href="#cite_note-72">[61]</a> This unphysical behavior, known as a "noncollision singularity",<a href="#cite_note-Barrow-Green2008-65">[54]</a> depends upon the masses being pointlike and able to approach one another arbitrarily closely, as well as the lack of a <a href="/wiki/Speed_of_light" title="Speed of light">relativistic speed limit</a> in Newtonian physics.<a href="#cite_note-73">[62]</a> It is not yet known whether or not the Euler and Navier–Stokes equations exhibit the analogous behavior of initially smooth solutions "blowing up" in finite time. The question of <a href="/wiki/Navier%E2%80%93Stokes_existence_and_smoothness" title="Navier–Stokes existence and smoothness">existence and smoothness of Navier–Stokes solutions</a> is one of the <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a>.<a href="#cite_note-74">[63]</a> <div class="mw-heading mw-heading2"><h2 id="Relation_to_other_formulations_of_classical_physics">Relation to other formulations of classical physics</h2></div> Classical mechanics can be mathematically formulated in multiple different ways, other than the "Newtonian" description (which itself, of course, incorporates contributions from others both before and after Newton). The physical content of these different formulations is the same as the Newtonian, but they provide different insights and facilitate different types of calculations. For example, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a> helps make apparent the connection between symmetries and conservation laws, and it is useful when calculating the motion of constrained bodies, like a mass restricted to move along a curving track or on the surface of a sphere.<a href="#cite_note-:2-21">[18]</a>: 48  <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a> is convenient for <a href="/wiki/Statistical_physics" class="mw-redirect" title="Statistical physics">statistical physics</a>,<a href="#cite_note-75">[64]</a><a href="#cite_note-:5-76">[65]</a>: 57  leads to further insight about symmetry,<a href="#cite_note-:2-21">[18]</a>: 251  and can be developed into sophisticated techniques for <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbation theory</a>.<a href="#cite_note-:2-21">[18]</a>: 284  Due to the breadth of these topics, the discussion here will be confined to concise treatments of how they reformulate Newton's laws of motion. <div class="mw-heading mw-heading3"><h3 id="Lagrangian">Lagrangian</h3></div> <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a> differs from the Newtonian formulation by considering entire trajectories at once rather than predicting a body's motion at a single instant.<a href="#cite_note-:2-21">[18]</a>: 109  It is traditional in Lagrangian mechanics to denote position with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> and velocity with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {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 {\dot {q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399dc6b6e91a780c89824ccc26b4453b289e4387" 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 {\dot {q}}}">. The simplest example is a massive point particle, the Lagrangian for which can be written as the difference between its kinetic and potential energies: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(q,{\dot {q}})=T-V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <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 stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo>−</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(q,{\dot {q}})=T-V,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d848e6f392a4be2e0335af8674afc81bc9cd2a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.882ex; height:2.843ex;" alt="{\displaystyle L(q,{\dot {q}})=T-V,}"> where the kinetic energy is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {1}{2}}m{\dot {q}}^{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> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {1}{2}}m{\dot {q}}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41425bea2f0e678155540035df918a9b88d24f0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.205ex; height:5.176ex;" alt="{\displaystyle T={\frac {1}{2}}m{\dot {q}}^{2}}"> and the potential energy is some function of the position, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(q)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dac6b17a8a7818fb4a89bd32f85daec2eb233f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.666ex; height:2.843ex;" alt="{\displaystyle V(q)}">. The physical path that the particle will take between an initial point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{i}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2752dcbff884354069fe332b8e51eb0a70a531b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.837ex; height:2.009ex;" alt="{\displaystyle q_{i}}"> and a final point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{f}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f2657d91f2f7644c467e3e49a30cfea9c18747" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.173ex; height:2.343ex;" alt="{\displaystyle q_{f}}"> is the path for which the integral of the Lagrangian is "stationary". That is, the physical path has the property that small perturbations of it will, to a first approximation, not change the integral of the Lagrangian. <a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a> provides the mathematical tools for finding this path.<a href="#cite_note-Boas-50">[42]</a>: 485  Applying the calculus of variations to the task of finding the path yields the <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equation</a> for the particle, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}}}\right)={\frac {\partial L}{\partial q}}.}"> <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> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</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>q</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}}}\right)={\frac {\partial L}{\partial q}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b9e7b66e2a9b32f0be390fd481d1d49cc35850" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.919ex; height:6.176ex;" alt="{\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}}}\right)={\frac {\partial L}{\partial q}}.}"> Evaluating the <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> of the Lagrangian gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}(m{\dot {q}})=-{\frac {dV}{dq}},}"> <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> <mo stretchy="false">(</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>V</mi> </mrow> <mrow> <mi>d</mi> <mi>q</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}(m{\dot {q}})=-{\frac {dV}{dq}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6630be7552161d69cf60b32e3b29c102528eae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.511ex; height:5.843ex;" alt="{\displaystyle {\frac {d}{dt}}(m{\dot {q}})=-{\frac {dV}{dq}},}"> which is a restatement of Newton's second law. The left-hand side is the time derivative of the momentum, and the right-hand side is the force, represented in terms of the potential energy.<a href="#cite_note-:1-11">[9]</a>: 737  <a href="/wiki/Course_of_Theoretical_Physics" title="Course of Theoretical Physics">Landau and Lifshitz</a> argue that the Lagrangian formulation makes the conceptual content of classical mechanics more clear than starting with Newton's laws.<a href="#cite_note-Landau-30">[26]</a> Lagrangian mechanics provides a convenient framework in which to prove <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a>, which relates symmetries and conservation laws.<a href="#cite_note-77">[66]</a> The conservation of momentum can be derived by applying Noether's theorem to a Lagrangian for a multi-particle system, and so, Newton's third law is a theorem rather than an assumption.<a href="#cite_note-:2-21">[18]</a>: 124  <div class="mw-heading mw-heading3"><h3 id="Hamiltonian">Hamiltonian</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Noether.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Noether.jpg/170px-Noether.jpg" decoding="async" width="170" height="259" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Noether.jpg/255px-Noether.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Noether.jpg/340px-Noether.jpg 2x" data-file-width="1208" data-file-height="1840" /></a><figcaption><a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a>, whose 1915 proof of <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">a celebrated theorem that relates symmetries and conservation laws</a> was a key development in modern physics and can be conveniently stated in the language of Lagrangian or Hamiltonian mechanics</figcaption></figure> In <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>, the dynamics of a system are represented by a function called the Hamiltonian, which in many cases of interest is equal to the total energy of the system.<a href="#cite_note-:1-11">[9]</a>: 742  The Hamiltonian is a function of the positions and the momenta of all the bodies making up the system, and it may also depend explicitly upon time. The time derivatives of the position and momentum variables are given by partial derivatives of the Hamiltonian, via <a href="/wiki/Hamilton%27s_equations" class="mw-redirect" title="Hamilton's equations">Hamilton's equations</a>.<a href="#cite_note-:2-21">[18]</a>: 203  The simplest example is a point mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> constrained to move in a straight line, under the effect of a potential. Writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> for the position coordinate and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> for the body's momentum, the Hamiltonian is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}(p,q)={\frac {p^{2}}{2m}}+V(q).}"> <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">H</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(p,q)={\frac {p^{2}}{2m}}+V(q).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec3c30ae260b4173db99919802aed70c4da0ade" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.337ex; height:5.676ex;" alt="{\displaystyle {\mathcal {H}}(p,q)={\frac {p^{2}}{2m}}+V(q).}"> In this example, Hamilton's equations are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dq}{dt}}={\frac {\partial {\mathcal {H}}}{\partial p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>q</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dq}{dt}}={\frac {\partial {\mathcal {H}}}{\partial p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca624fc3fc78b26d00a28e67ea897fb07b5bd1c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.338ex; height:5.843ex;" alt="{\displaystyle {\frac {dq}{dt}}={\frac {\partial {\mathcal {H}}}{\partial p}}}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dp}{dt}}=-{\frac {\partial {\mathcal {H}}}{\partial q}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>q</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dp}{dt}}=-{\frac {\partial {\mathcal {H}}}{\partial q}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e1f128982c8332b653cf0b1982308b3dfc61b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.893ex; height:5.843ex;" alt="{\displaystyle {\frac {dp}{dt}}=-{\frac {\partial {\mathcal {H}}}{\partial q}}.}"> Evaluating these partial derivatives, the former equation becomes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dq}{dt}}={\frac {p}{m}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>q</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>m</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dq}{dt}}={\frac {p}{m}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8747695fd4a3b4722410a4084b77aee8bf15ce18" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.743ex; height:5.509ex;" alt="{\displaystyle {\frac {dq}{dt}}={\frac {p}{m}},}"> which reproduces the familiar statement that a body's momentum is the product of its mass and velocity. The time derivative of the momentum is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dp}{dt}}=-{\frac {dV}{dq}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>V</mi> </mrow> <mrow> <mi>d</mi> <mi>q</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dp}{dt}}=-{\frac {dV}{dq}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a838a49c1d0d0862911e3d00296d7cb6266641" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.614ex; height:5.843ex;" alt="{\displaystyle {\frac {dp}{dt}}=-{\frac {dV}{dq}},}"> which, upon identifying the negative derivative of the potential with the force, is just Newton's second law once again.<a href="#cite_note-:3-68">[57]</a><a href="#cite_note-:1-11">[9]</a>: 742  As in the Lagrangian formulation, in Hamiltonian mechanics the conservation of momentum can be derived using Noether's theorem, making Newton's third law an idea that is deduced rather than assumed.<a href="#cite_note-:2-21">[18]</a>: 251  Among the proposals to reform the standard introductory-physics curriculum is one that teaches the concept of energy before that of force, essentially "introductory Hamiltonian mechanics".<a href="#cite_note-78">[67]</a><a href="#cite_note-79">[68]</a> <div class="mw-heading mw-heading3"><h3 id="Hamilton–Jacobi">Hamilton–Jacobi</h3></div> The <a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a> provides yet another formulation of classical mechanics, one which makes it mathematically analogous to <a href="/wiki/Wave_optics" class="mw-redirect" title="Wave optics">wave optics</a>.<a href="#cite_note-:2-21">[18]</a>: 284 <a href="#cite_note-80">[69]</a> This formulation also uses Hamiltonian functions, but in a different way than the formulation described above. The paths taken by bodies or collections of bodies are deduced from a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {q} _{1},\mathbf {q} _{2},\ldots ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} _{1},\mathbf {q} _{2},\ldots ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78b62f7d77a6133186ad5514a08d639405f05794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.291ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {q} _{1},\mathbf {q} _{2},\ldots ,t)}"> of positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {q} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ce6be9c9335b20681f1b784557c574a69f28e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.211ex; height:2.176ex;" alt="{\displaystyle \mathbf {q} _{i}}"> and time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">. The Hamiltonian is incorporated into the Hamilton–Jacobi equation, a <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">. Bodies move over time in such a way that their trajectories are perpendicular to the surfaces of constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, analogously to how a light ray propagates in the direction perpendicular to its wavefront. This is simplest to express for the case of a single point mass, in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {q} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a9798e69ae297664214c3045d8bd0b1c0208d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.598ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {q} ,t)}">, and the point mass moves in the direction along which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> changes most steeply. In other words, the momentum of the point mass is the <a href="/wiki/Gradient" title="Gradient">gradient</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} ={\frac {1}{m}}\mathbf {\nabla } S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\frac {1}{m}}\mathbf {\nabla } S.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bf70cd53a9f7756425f53e7d58d32e765be48e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.468ex; height:5.176ex;" alt="{\displaystyle \mathbf {v} ={\frac {1}{m}}\mathbf {\nabla } S.}"> The Hamilton–Jacobi equation for a point mass is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\partial S}{\partial t}}=H\left(\mathbf {q} ,\mathbf {\nabla } S,t\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>S</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\partial S}{\partial t}}=H\left(\mathbf {q} ,\mathbf {\nabla } S,t\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a146b7c7f7e666d8195ebf60483f9415eaaf4191" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.612ex; height:5.509ex;" alt="{\displaystyle -{\frac {\partial S}{\partial t}}=H\left(\mathbf {q} ,\mathbf {\nabla } S,t\right).}"> The relation to Newton's laws can be seen by considering a point mass moving in a time-independent potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <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 V(\mathbf {q} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae9b96c4a6f3cfc2aa2c2a878ca943b300b70e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.012ex; height:2.843ex;" alt="{\displaystyle V(\mathbf {q} )}">, in which case the Hamilton–Jacobi equation becomes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\partial S}{\partial t}}={\frac {1}{2m}}\left(\mathbf {\nabla } S\right)^{2}+V(\mathbf {q} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>S</mi> </mrow> <mo>)</mo> </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"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\partial S}{\partial t}}={\frac {1}{2m}}\left(\mathbf {\nabla } S\right)^{2}+V(\mathbf {q} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1170cbf9e33f388936133349b42f7c7c8135e62c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.397ex; height:5.509ex;" alt="{\displaystyle -{\frac {\partial S}{\partial t}}={\frac {1}{2m}}\left(\mathbf {\nabla } S\right)^{2}+V(\mathbf {q} ).}"> Taking the gradient of both sides, this becomes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathbf {\nabla } {\frac {\partial S}{\partial t}}={\frac {1}{2m}}\mathbf {\nabla } \left(\mathbf {\nabla } S\right)^{2}+\mathbf {\nabla } V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mathbf {\nabla } {\frac {\partial S}{\partial t}}={\frac {1}{2m}}\mathbf {\nabla } \left(\mathbf {\nabla } S\right)^{2}+\mathbf {\nabla } V.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e69588e25e8af99c9fbf883ccf13c9b583f8391" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.98ex; height:5.509ex;" alt="{\displaystyle -\mathbf {\nabla } {\frac {\partial S}{\partial t}}={\frac {1}{2m}}\mathbf {\nabla } \left(\mathbf {\nabla } S\right)^{2}+\mathbf {\nabla } V.}"> Interchanging the order of the partial derivatives on the left-hand side, and using the <a href="/wiki/Power_rule" title="Power rule">power</a> and <a href="/wiki/Chain_rule" title="Chain rule">chain rules</a> on the first term on the right-hand side, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\partial }{\partial t}}\mathbf {\nabla } S={\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\mathbf {\nabla } S+\mathbf {\nabla } V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>S</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>S</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\partial }{\partial t}}\mathbf {\nabla } S={\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\mathbf {\nabla } S+\mathbf {\nabla } V.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11ac208298a72a39b478f82e3477b0e32b98d474" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.491ex; height:5.509ex;" alt="{\displaystyle -{\frac {\partial }{\partial t}}\mathbf {\nabla } S={\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\mathbf {\nabla } S+\mathbf {\nabla } V.}"> Gathering together the terms that depend upon the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\frac {\partial }{\partial t}}+{\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\right]\mathbf {\nabla } S=-\mathbf {\nabla } V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>S</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>S</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\frac {\partial }{\partial t}}+{\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\right]\mathbf {\nabla } S=-\mathbf {\nabla } V.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9e3cf63a74bdfb6e4ac08c85c31828ee89fb07" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.511ex; height:6.176ex;" alt="{\displaystyle \left[{\frac {\partial }{\partial t}}+{\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\right]\mathbf {\nabla } S=-\mathbf {\nabla } V.}"> This is another re-expression of Newton's second law.<a href="#cite_note-81">[70]</a> The expression in brackets is a <a href="/wiki/Material_derivative" title="Material derivative">total or material derivative</a> as mentioned above,<a href="#cite_note-82">[71]</a> in which the first term indicates how the function being differentiated changes over time at a fixed location, and the second term captures how a moving particle will see different values of that function as it travels from place to place: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\frac {\partial }{\partial t}}+{\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\right]=\left[{\frac {\partial }{\partial t}}+\mathbf {v} \cdot \mathbf {\nabla } \right]={\frac {d}{dt}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>S</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\frac {\partial }{\partial t}}+{\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\right]=\left[{\frac {\partial }{\partial t}}+\mathbf {v} \cdot \mathbf {\nabla } \right]={\frac {d}{dt}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba77978a485bdd9eafdda5aa195e2f828c16863" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.463ex; height:6.176ex;" alt="{\displaystyle \left[{\frac {\partial }{\partial t}}+{\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\right]=\left[{\frac {\partial }{\partial t}}+\mathbf {v} \cdot \mathbf {\nabla } \right]={\frac {d}{dt}}.}"> <div class="mw-heading mw-heading2"><h2 id="Relation_to_other_physical_theories">Relation to other physical theories</h2></div> <div class="mw-heading mw-heading3"><h3 id="Thermodynamics_and_statistical_physics">Thermodynamics and statistical physics</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Brownian_motion_large.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Brownian_motion_large.gif/220px-Brownian_motion_large.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c2/Brownian_motion_large.gif 1.5x" data-file-width="240" data-file-height="240" /></a><figcaption>A simulation of a larger, but still microscopic, particle (in yellow) surrounded by a gas of smaller particles, illustrating <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a></figcaption></figure> In <a href="/wiki/Statistical_physics" class="mw-redirect" title="Statistical physics">statistical physics</a>, the <a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory of gases</a> applies Newton's laws of motion to large numbers (typically on the order of the <a href="/wiki/Avogadro_constant" title="Avogadro constant">Avogadro number</a>) of particles. Kinetic theory can explain, for example, the <a href="/wiki/Pressure" title="Pressure">pressure</a> that a gas exerts upon the container holding it as the aggregate of many impacts of atoms, each imparting a tiny amount of momentum.<a href="#cite_note-:5-76">[65]</a>: 62  The <a href="/wiki/Langevin_equation" title="Langevin equation">Langevin equation</a> is a special case of Newton's second law, adapted for the case of describing a small object bombarded stochastically by even smaller ones.<a href="#cite_note-:4-83">[72]</a>: 235  It can be written<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\mathbf {a} =-\gamma \mathbf {v} +\mathbf {\xi } \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\mathbf {a} =-\gamma \mathbf {v} +\mathbf {\xi } \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4701c269fde542fa22d40bca66e70ad383f8ae2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.177ex; height:2.676ex;" alt="{\displaystyle m\mathbf {a} =-\gamma \mathbf {v} +\mathbf {\xi } \,}">where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"> is a <a href="/wiki/Drag_coefficient" title="Drag coefficient">drag coefficient</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\xi } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\xi } }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e2a51fb5522ad140f19673a76daef706190448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \mathbf {\xi } }"> is a force that varies randomly from instant to instant, representing the net effect of collisions with the surrounding particles. This is used to model <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a>.<a href="#cite_note-84">[73]</a> <div class="mw-heading mw-heading3"><h3 id="Electromagnetism">Electromagnetism</h3></div> Newton's three laws can be applied to phenomena involving <a href="/wiki/Electricity" title="Electricity">electricity</a> and <a href="/wiki/Magnetism" title="Magnetism">magnetism</a>, though subtleties and caveats exist. <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</a> for the electric force between two stationary, <a href="/wiki/Electric_charge" title="Electric charge">electrically charged</a> bodies has much the same mathematical form as Newton's law of universal gravitation: the force is proportional to the product of the charges, inversely proportional to the square of the distance between them, and directed along the straight line between them. The Coulomb force that a charge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9daa41f6e8f78ea6bb5711d7ac97901ce564b94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{1}}"> exerts upon a charge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2d05084feb02b8ba29b0673440fb673b102589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{2}}"> is equal in magnitude to the force that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2d05084feb02b8ba29b0673440fb673b102589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{2}}"> exerts upon <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9daa41f6e8f78ea6bb5711d7ac97901ce564b94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{1}}">, and it points in the exact opposite direction. Coulomb's law is thus consistent with Newton's third law.<a href="#cite_note-85">[74]</a> Electromagnetism treats forces as produced by fields acting upon charges. The <a href="/wiki/Lorentz_force_law" class="mw-redirect" title="Lorentz force law">Lorentz force law</a> provides an expression for the force upon a charged body that can be plugged into Newton's second law in order to calculate its acceleration.<a href="#cite_note-86">[75]</a>: 85  According to the Lorentz force law, a charged body in an electric field experiences a force in the direction of that field, a force proportional to its charge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> and to the strength of the electric field. In addition, a moving charged body in a magnetic field experiences a force that is also proportional to its charge, in a direction perpendicular to both the field and the body's direction of motion. Using the vector <a href="/wiki/Cross_product" title="Cross product">cross product</a>,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B} .}"> <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>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6897d76dd298da4a1aaeb6ef7175661d7589e6ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.317ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B} .}"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cyclotron_motion.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Cyclotron_motion.jpg/220px-Cyclotron_motion.jpg" decoding="async" width="220" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Cyclotron_motion.jpg/330px-Cyclotron_motion.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Cyclotron_motion.jpg/440px-Cyclotron_motion.jpg 2x" data-file-width="2632" data-file-height="1866" /></a><figcaption>The Lorentz force law in effect: electrons are bent into a circular trajectory by a magnetic field.</figcaption></figure>If the electric field vanishes (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fac8fccc5dc45d85848483c2bc6b77b4b7033e0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.018ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} =0}">), then the force will be perpendicular to the charge's motion, just as in the case of uniform circular motion studied above, and the charge will circle (or more generally move in a <a href="/wiki/Helix" title="Helix">helix</a>) around the magnetic field lines at the <a href="/wiki/Cyclotron_resonance" title="Cyclotron resonance">cyclotron frequency</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =qB/m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>q</mi> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =qB/m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b49dfeea500cc3c3fe955713b88bc3354233e7e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.581ex; height:2.843ex;" alt="{\displaystyle \omega =qB/m}">.<a href="#cite_note-:4-83">[72]</a>: 222  <a href="/wiki/Mass_spectrometry" title="Mass spectrometry">Mass spectrometry</a> works by applying electric and/or magnetic fields to moving charges and measuring the resulting acceleration, which by the Lorentz force law yields the <a href="/wiki/Mass-to-charge_ratio" title="Mass-to-charge ratio">mass-to-charge ratio</a>.<a href="#cite_note-87">[76]</a> Collections of charged bodies do not always obey Newton's third law: there can be a change of one body's momentum without a compensatory change in the momentum of another. The discrepancy is accounted for by momentum carried by the electromagnetic field itself. The momentum per unit volume of the electromagnetic field is proportional to the <a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a>.<a href="#cite_note-Panofsky1962-88">[77]</a>: 184 <a href="#cite_note-89">[78]</a> There is subtle conceptual conflict between electromagnetism and Newton's first law: <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's theory of electromagnetism</a> predicts that electromagnetic waves will travel through empty space at a constant, definite speed. Thus, some inertial observers seemingly have a privileged status over the others, namely those who measure the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> and find it to be the value predicted by the Maxwell equations. In other words, light provides an absolute standard for speed, yet the principle of inertia holds that there should be no such standard. This tension is resolved in the theory of special relativity, which revises the notions of space and time in such a way that all inertial observers will agree upon the speed of light in vacuum.<a href="#cite_note-92">[note 12]</a> <div class="mw-heading mw-heading3"><h3 id="Special_relativity">Special relativity</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Relativistic_mechanics" title="Relativistic mechanics">Relativistic mechanics</a> and <a href="/wiki/Acceleration_(special_relativity)" title="Acceleration (special relativity)">Acceleration (special relativity)</a></div> In special relativity, the rule that Wilczek called "Newton's Zeroth Law" breaks down: the mass of a composite object is not merely the sum of the masses of the individual pieces.<a href="#cite_note-:7-93">[81]</a>: 33  Newton's first law, inertial motion, remains true. A form of Newton's second law, that force is the rate of change of momentum, also holds, as does the conservation of momentum. However, the definition of momentum is modified. Among the consequences of this is the fact that the more quickly a body moves, the harder it is to accelerate, and so, no matter how much force is applied, a body cannot be accelerated to the speed of light. Depending on the problem at hand, momentum in special relativity can be represented as a three-dimensional vector, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =m\gamma \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =m\gamma \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/212dc9b9ce048465636309e9442073e1c1ce0b27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.297ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} =m\gamma \mathbf {v} }">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> is the body's <a href="/wiki/Rest_mass" class="mw-redirect" title="Rest mass">rest mass</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"> is the <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a>, which depends upon the body's speed. Alternatively, momentum and force can be represented as <a href="/wiki/Four-vector" title="Four-vector">four-vectors</a>.<a href="#cite_note-94">[82]</a>: 107  Newton's third law must be modified in special relativity. The third law refers to the forces between two bodies at the same moment in time, and a key feature of special relativity is that simultaneity is relative. Events that happen at the same time relative to one observer can happen at different times relative to another. So, in a given observer's frame of reference, action and reaction may not be exactly opposite, and the total momentum of interacting bodies may not be conserved. The conservation of momentum is restored by including the momentum stored in the field that describes the bodies' interaction.<a href="#cite_note-95">[83]</a><a href="#cite_note-96">[84]</a> Newtonian mechanics is a good approximation to special relativity when the speeds involved are small compared to that of light.<a href="#cite_note-97">[85]</a>: 131  <div class="mw-heading mw-heading3"><h3 id="General_relativity">General relativity</h3></div> <a href="/wiki/General_relativity" title="General relativity">General relativity</a> is a theory of gravity that advances beyond that of Newton. In general relativity, the gravitational force of Newtonian mechanics is reimagined as curvature of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. A curved path like an orbit, attributed to a gravitational force in Newtonian mechanics, is not the result of a force deflecting a body from an ideal straight-line path, but rather the body's attempt to fall freely through a background that is itself curved by the presence of other masses. A remark by <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a> that has become proverbial among physicists summarizes the theory: "Spacetime tells matter how to move; matter tells spacetime how to curve."<a href="#cite_note-Wheeler-98">[86]</a><a href="#cite_note-99">[87]</a> Wheeler himself thought of this reciprocal relationship as a modern, generalized form of Newton's third law.<a href="#cite_note-Wheeler-98">[86]</a> The relation between matter distribution and spacetime curvature is given by the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a>, which require <a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">tensor calculus</a> to express.<a href="#cite_note-:7-93">[81]</a>: 43 <a href="#cite_note-100">[88]</a> The Newtonian theory of gravity is a good approximation to the predictions of general relativity when gravitational effects are weak and objects are moving slowly compared to the speed of light.<a href="#cite_note-:6-90">[79]</a>: 327 <a href="#cite_note-101">[89]</a> <div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3></div> <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is <a href="/wiki/Bell%27s_theorem" title="Bell's theorem">very different from that of classical physics</a>. Instead of thinking about quantities like position, momentum, and energy as properties that an object has, one considers what result might appear when a <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">measurement</a> of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result.<a href="#cite_note-102">[90]</a><a href="#cite_note-103">[91]</a> The <a href="/wiki/Expectation_value_(quantum_mechanics)" title="Expectation value (quantum mechanics)">expectation value</a> for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.<a href="#cite_note-104">[92]</a> The <a href="/wiki/Ehrenfest_theorem" title="Ehrenfest theorem">Ehrenfest theorem</a> provides a connection between quantum expectation values and Newton's second law, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, position and momentum are represented by mathematical entities known as <a href="/wiki/Hermitian_operator" class="mw-redirect" title="Hermitian operator">Hermitian operators</a>, and the <a href="/wiki/Born_rule" title="Born rule">Born rule</a> is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.<a href="#cite_note-107">[note 13]</a> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <style data-mw-deduplicate="TemplateStyles:r1248256098">@media all and (max-width:720px){.mw-parser-output .mod-gallery{width:100%!important}}.mw-parser-output .mod-gallery{display:table}.mw-parser-output .mod-gallery-default{background:transparent;margin-top:4px}.mw-parser-output .mod-gallery-center{margin-left:auto;margin-right:auto}.mw-parser-output .mod-gallery-left{float:left}.mw-parser-output .mod-gallery-right{float:right}.mw-parser-output .mod-gallery-none{float:none}.mw-parser-output .mod-gallery-collapsible{width:100%}.mw-parser-output .mod-gallery .title,.mw-parser-output .mod-gallery .main,.mw-parser-output .mod-gallery .footer{display:table-row}.mw-parser-output .mod-gallery .title>div{display:table-cell;padding:0 4px 4px;text-align:center;font-weight:bold}.mw-parser-output .mod-gallery .main>div{display:table-cell}.mw-parser-output .mod-gallery .gallery{line-height:1.35em}.mw-parser-output .mod-gallery .footer>div{display:table-cell;padding:4px;text-align:right;font-size:85%;line-height:1em}.mw-parser-output .mod-gallery .title>div *,.mw-parser-output .mod-gallery .footer>div *{overflow:visible}.mw-parser-output .mod-gallery .gallerybox img{background:none!important}.mw-parser-output .mod-gallery .bordered-images .thumb img{border:solid var(--background-color-neutral,#eaecf0)1px}.mw-parser-output .mod-gallery .whitebg .thumb{background:var(--background-color-base,#fff)!important}</style><div class="mod-gallery mod-gallery-default mod-gallery-center"><div class="main"><div><ul class="gallery mw-gallery-traditional nochecker bordered-images whitebg"> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><a href="/wiki/File:Portrait_of_Sir_Isaac_Newton,_1689.jpg" class="mw-file-description" title="Isaac Newton (1643–1727), in a 1689 portrait by Godfrey Kneller"><img alt="Isaac Newton (1643–1727), in a 1689 portrait by Godfrey Kneller" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Portrait_of_Sir_Isaac_Newton%2C_1689.jpg/149px-Portrait_of_Sir_Isaac_Newton%2C_1689.jpg" decoding="async" width="149" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Portrait_of_Sir_Isaac_Newton%2C_1689.jpg/224px-Portrait_of_Sir_Isaac_Newton%2C_1689.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Portrait_of_Sir_Isaac_Newton%2C_1689.jpg/299px-Portrait_of_Sir_Isaac_Newton%2C_1689.jpg 2x" data-file-width="2218" data-file-height="2671" /></a></div> <div class="gallerytext">Isaac Newton (1643–1727), in a 1689 portrait by <a href="/wiki/Godfrey_Kneller" title="Godfrey Kneller">Godfrey Kneller</a></div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><a href="/wiki/File:NewtonsPrincipia.jpg" class="mw-file-description" title="Newton's own copy of his Principia, with hand-written corrections for the second edition, in the Wren Library at Trinity College, Cambridge"><img alt="Newton's own copy of his Principia, with hand-written corrections for the second edition, in the Wren Library at Trinity College, Cambridge" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/NewtonsPrincipia.jpg/180px-NewtonsPrincipia.jpg" decoding="async" width="180" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/NewtonsPrincipia.jpg/270px-NewtonsPrincipia.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/NewtonsPrincipia.jpg/360px-NewtonsPrincipia.jpg 2x" data-file-width="1266" data-file-height="842" /></a></div> <div class="gallerytext">Newton's own copy of his <a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a>, with hand-written corrections for the second edition, in the <a href="/wiki/Wren_Library" title="Wren Library">Wren Library</a> at <a href="/wiki/Trinity_College,_Cambridge" title="Trinity College, Cambridge">Trinity College, Cambridge</a></div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><a href="/wiki/File:Newtons_laws_in_latin.jpg" class="mw-file-description" title="Newton's first and second laws, in Latin, from the original 1687 Principia Mathematica"><img alt="Newton's first and second laws, in Latin, from the original 1687 Principia Mathematica" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Newtons_laws_in_latin.jpg/115px-Newtons_laws_in_latin.jpg" decoding="async" width="115" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Newtons_laws_in_latin.jpg/173px-Newtons_laws_in_latin.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Newtons_laws_in_latin.jpg/231px-Newtons_laws_in_latin.jpg 2x" data-file-width="296" data-file-height="461" /></a></div> <div class="gallerytext">Newton's first and second laws, in Latin, from the original 1687 <a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia Mathematica</a></div> </li> </ul></div></div></div> The concepts invoked in Newton's laws of motion — mass, velocity, momentum, force — have predecessors in earlier work, and the content of Newtonian physics was further developed after Newton's time. Newton combined knowledge of celestial motions with the study of events on Earth and showed that one theory of mechanics could encompass both.<a href="#cite_note-110">[note 14]</a> <div class="mw-heading mw-heading3"><h3 id="Antiquity_and_medieval_background">Antiquity and medieval background</h3></div> <div class="mw-heading mw-heading4"><h4 id="Aristotle_and_"violent"_motion">Aristotle and "violent" motion</h4></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Statue_at_the_Aristotle_University_of_Thessaloniki_(cropped).jpg" class="mw-file-description"><img alt="Statue of Aristotle" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Statue_at_the_Aristotle_University_of_Thessaloniki_%28cropped%29.jpg/150px-Statue_at_the_Aristotle_University_of_Thessaloniki_%28cropped%29.jpg" decoding="async" width="150" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Statue_at_the_Aristotle_University_of_Thessaloniki_%28cropped%29.jpg/226px-Statue_at_the_Aristotle_University_of_Thessaloniki_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Statue_at_the_Aristotle_University_of_Thessaloniki_%28cropped%29.jpg/300px-Statue_at_the_Aristotle_University_of_Thessaloniki_%28cropped%29.jpg 2x" data-file-width="1935" data-file-height="2641" /></a><figcaption>Aristotle (384–322 <a href="/wiki/BCE" class="mw-redirect" title="BCE">BCE</a>)</figcaption></figure> The subject of physics is often traced back to <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, but the history of the concepts involved is obscured by multiple factors. An exact correspondence between Aristotelian and modern concepts is not simple to establish: Aristotle did not clearly distinguish what we would call speed and force, used the same term for <a href="/wiki/Density" title="Density">density</a> and <a href="/wiki/Viscosity" title="Viscosity">viscosity</a>, and conceived of motion as always through a medium, rather than through space. In addition, some concepts often termed "Aristotelian" might better be attributed to his followers and commentators upon him.<a href="#cite_note-111">[97]</a> These commentators found that Aristotelian physics had difficulty explaining projectile motion.<a href="#cite_note-113">[note 15]</a> Aristotle divided motion into two types: "natural" and "violent". The "natural" motion of terrestrial solid matter was to fall downwards, whereas a "violent" motion could push a body sideways. Moreover, in Aristotelian physics, a "violent" motion requires an immediate cause; separated from the cause of its "violent" motion, a body would revert to its "natural" behavior. Yet, a javelin continues moving after it leaves the thrower's hand. Aristotle concluded that the air around the javelin must be imparted with the ability to move the javelin forward. <div class="mw-heading mw-heading4"><h4 id="Philoponus_and_impetus">Philoponus and impetus</h4></div> <a href="/wiki/John_Philoponus" title="John Philoponus">John Philoponus</a>, a <a href="/wiki/Byzantine_Greek" class="mw-redirect" title="Byzantine Greek">Byzantine Greek</a> thinker active during the sixth century, found this absurd: the same medium, air, was somehow responsible both for sustaining motion and for impeding it. If Aristotle's idea were true, Philoponus said, armies would launch weapons by blowing upon them with bellows. Philoponus argued that setting a body into motion imparted a quality, <a href="/wiki/Theory_of_impetus" title="Theory of impetus">impetus</a>, that would be contained within the body itself. As long as its impetus was sustained, the body would continue to move.<a href="#cite_note-114">[99]</a>: 47  In the following centuries, versions of impetus theory were advanced by individuals including <a href="/wiki/Nur_ad-Din_al-Bitruji" title="Nur ad-Din al-Bitruji">Nur ad-Din al-Bitruji</a>, <a href="/wiki/Avicenna" title="Avicenna">Avicenna</a>, <a href="/wiki/Abu%27l-Barak%C4%81t_al-Baghd%C4%81d%C4%AB" class="mw-redirect" title="Abu'l-Barakāt al-Baghdādī">Abu'l-Barakāt al-Baghdādī</a>, <a href="/wiki/John_Buridan" class="mw-redirect" title="John Buridan">John Buridan</a>, and <a href="/wiki/Albert_of_Saxony_(philosopher)" title="Albert of Saxony (philosopher)">Albert of Saxony</a>. In retrospect, the idea of impetus can be seen as a forerunner of the modern concept of momentum.<a href="#cite_note-116">[note 16]</a> The intuition that objects move according to some kind of impetus persists in many students of introductory physics.<a href="#cite_note-117">[101]</a> <div class="mw-heading mw-heading3"><h3 id="Inertia_and_the_first_law">Inertia and the first law</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Galileo_Galilei#Inertia" title="Galileo Galilei">Galileo Galilei § Inertia</a></div> The French philosopher <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> introduced the concept of inertia by way of his "laws of nature" in <a href="/wiki/The_World_(book)" title="The World (book)">The World</a> (Traité du monde et de la lumière) written 1629–33. However, The World purported a <a href="/wiki/Heliocentrism" title="Heliocentrism">heliocentric</a> worldview, and in 1633 this view had given rise a great conflict between <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a> and the <a href="/wiki/Roman_Inquisition" title="Roman Inquisition">Roman Catholic Inquisition</a>. Descartes knew about this controversy and did not wish to get involved. The World was not published until 1664, ten years after his death.<a href="#cite_note-:8-118">[102]</a> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Galileo.arp.300pix.jpg" class="mw-file-description"><img alt="Justus Sustermans - Portrait of Galileo Galilei" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Galileo.arp.300pix.jpg/150px-Galileo.arp.300pix.jpg" decoding="async" width="150" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Galileo.arp.300pix.jpg/225px-Galileo.arp.300pix.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Galileo.arp.300pix.jpg/300px-Galileo.arp.300pix.jpg 2x" data-file-width="1180" data-file-height="1448" /></a><figcaption>Galileo Galilei (1564–1642)</figcaption></figure> The modern concept of inertia is credited to Galileo. Based on his experiments, Galileo concluded that the "natural" behavior of a moving body was to keep moving, until something else interfered with it. In Two New Sciences (1638) Galileo wrote:<a href="#cite_note-119">[103]</a><a href="#cite_note-120">[104]</a><style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">Imagine any particle projected along a horizontal plane without friction; then we know, from what has been more fully explained in the preceding pages, that this particle will move along this same plane with a motion which is uniform and perpetual, provided the plane has no limits.</blockquote><figure typeof="mw:File/Thumb"><a href="/wiki/File:Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_(cropped)2.jpg" class="mw-file-description"><img alt="Portrait of René Descartes" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg/150px-Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg" decoding="async" width="150" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg/225px-Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg/299px-Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg 2x" data-file-width="478" data-file-height="489" /></a><figcaption>René Descartes (1596–1650)</figcaption></figure>Galileo recognized that in projectile motion, the Earth's gravity affects vertical but not horizontal motion.<a href="#cite_note-121">[105]</a> However, Galileo's idea of inertia was not exactly the one that would be codified into Newton's first law. Galileo thought that a body moving a long distance inertially would follow the curve of the Earth. This idea was corrected by <a href="/wiki/Isaac_Beeckman" title="Isaac Beeckman">Isaac Beeckman</a>, Descartes, and <a href="/wiki/Pierre_Gassendi" title="Pierre Gassendi">Pierre Gassendi</a>, who recognized that inertial motion should be motion in a straight line.<a href="#cite_note-122">[106]</a> Descartes published his laws of nature (laws of motion) with this correction in <a href="/wiki/Principles_of_Philosophy" title="Principles of Philosophy">Principles of Philosophy</a> (Principia Philosophiae) in 1644, with the heliocentric part toned down.<a href="#cite_note-:02-123">[107]</a><a href="#cite_note-:8-118">[102]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Breaking_String.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Breaking_String.PNG/220px-Breaking_String.PNG" decoding="async" width="220" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Breaking_String.PNG/330px-Breaking_String.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Breaking_String.PNG/440px-Breaking_String.PNG 2x" data-file-width="933" data-file-height="613" /></a><figcaption>Ball in circular motion has string cut and flies off tangentially.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">First Law of Nature: Each thing when left to itself continues in the same state; so any moving body goes on moving until something stops it.</blockquote><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">Second Law of Nature: Each moving thing if left to itself moves in a straight line; so any body moving in a circle always tends to move away from the centre of the circle.</blockquote> According to American philosopher <a href="/wiki/Richard_J._Blackwell" title="Richard J. Blackwell">Richard J. Blackwell</a>, Dutch scientist <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> had worked out his own, concise version of the law in 1656.<a href="#cite_note-:9-124">[108]</a> It was not published until 1703, eight years after his death, in the opening paragraph of De Motu Corporum ex Percussione. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">Hypothesis I: Any body already in motion will continue to move perpetually with the same speed and in a straight line unless it is impeded.</blockquote> According to Huygens, this law was already known by Galileo and Descartes among others.<a href="#cite_note-:9-124">[108]</a> <div class="mw-heading mw-heading3"><h3 id="Force_and_the_second_law">Force and the second law</h3></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Christiaan_Huygens-painting_(cropped).jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Christiaan_Huygens-painting_%28cropped%29.jpeg/150px-Christiaan_Huygens-painting_%28cropped%29.jpeg" decoding="async" width="150" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Christiaan_Huygens-painting_%28cropped%29.jpeg/226px-Christiaan_Huygens-painting_%28cropped%29.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Christiaan_Huygens-painting_%28cropped%29.jpeg/301px-Christiaan_Huygens-painting_%28cropped%29.jpeg 2x" data-file-width="765" data-file-height="789" /></a><figcaption>Christiaan Huygens (1629–1695)</figcaption></figure> Christiaan Huygens, in his <a href="/wiki/Horologium_Oscillatorium" title="Horologium Oscillatorium">Horologium Oscillatorium</a> (1673), put forth the hypothesis that "By the action of gravity, whatever its sources, it happens that bodies are moved by a motion composed both of a uniform motion in one direction or another and of a motion downward due to gravity." Newton's second law generalized this hypothesis from gravity to all forces.<a href="#cite_note-125">[109]</a> One important characteristic of Newtonian physics is that forces can <a href="/wiki/Action_at_a_distance" title="Action at a distance">act at a distance</a> without requiring physical contact.<a href="#cite_note-132">[note 17]</a> For example, the Sun and the Earth pull on each other gravitationally, despite being separated by millions of kilometres. This contrasts with the idea, championed by Descartes among others, that the Sun's gravity held planets in orbit by swirling them in a vortex of transparent matter, <a href="/wiki/Aether_theories" title="Aether theories">aether</a>.<a href="#cite_note-133">[116]</a> Newton considered aetherial explanations of force but ultimately rejected them.<a href="#cite_note-Newman2016-130">[114]</a> The study of magnetism by <a href="/wiki/William_Gilbert_(physician)" class="mw-redirect" title="William Gilbert (physician)">William Gilbert</a> and others created a precedent for thinking of immaterial forces,<a href="#cite_note-Newman2016-130">[114]</a> and unable to find a quantitatively satisfactory explanation of his law of gravity in terms of an aetherial model, Newton eventually declared, "<a href="/wiki/Hypotheses_non_fingo" title="Hypotheses non fingo">I feign no hypotheses</a>": whether or not a model like Descartes's vortices could be found to underlie the Principia's theories of motion and gravity, the first grounds for judging them must be the successful predictions they made.<a href="#cite_note-134">[117]</a> And indeed, since Newton's time <a href="/wiki/Mechanical_explanations_of_gravitation" title="Mechanical explanations of gravitation">every attempt at such a model has failed</a>. <div class="mw-heading mw-heading3"><h3 id="Momentum_conservation_and_the_third_law">Momentum conservation and the third law</h3></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:JKepler_(cropped).jpg" class="mw-file-description"><img alt="Portrait of Johannes Kepler" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/JKepler_%28cropped%29.jpg/150px-JKepler_%28cropped%29.jpg" decoding="async" width="150" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/JKepler_%28cropped%29.jpg/226px-JKepler_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/JKepler_%28cropped%29.jpg/300px-JKepler_%28cropped%29.jpg 2x" data-file-width="385" data-file-height="387" /></a><figcaption>Johannes Kepler (1571–1630)</figcaption></figure> <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> suggested that gravitational attractions were reciprocal — that, for example, the Moon pulls on the Earth while the Earth pulls on the Moon — but he did not argue that such pairs are equal and opposite.<a href="#cite_note-135">[118]</a> In his <a href="/wiki/Principles_of_Philosophy" title="Principles of Philosophy">Principles of Philosophy</a> (1644), Descartes introduced the idea that during a collision between bodies, a "quantity of motion" remains unchanged. Descartes defined this quantity somewhat imprecisely by adding up the products of the speed and "size" of each body, where "size" for him incorporated both volume and surface area.<a href="#cite_note-136">[119]</a> Moreover, Descartes thought of the universe as a <a href="/wiki/Plenum_(physics)" class="mw-redirect" title="Plenum (physics)">plenum</a>, that is, filled with matter, so all motion required a body to displace a medium as it moved. During the 1650s, Huygens studied collisions between hard spheres and deduced a principle that is now identified as the conservation of momentum.<a href="#cite_note-137">[120]</a><a href="#cite_note-138">[121]</a> <a href="/wiki/Christopher_Wren" title="Christopher Wren">Christopher Wren</a> would later deduce the same rules for <a href="/wiki/Elastic_collision" title="Elastic collision">elastic collisions</a> that Huygens had, and <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a> would apply momentum conservation to study <a href="/wiki/Inelastic_collision" title="Inelastic collision">inelastic collisions</a>. Newton cited the work of Huygens, Wren, and Wallis to support the validity of his third law.<a href="#cite_note-139">[122]</a> Newton arrived at his set of three laws incrementally. In a <a href="/wiki/De_motu_corporum_in_gyrum" title="De motu corporum in gyrum">1684 manuscript written to Huygens</a>, he listed four laws: the principle of inertia, the change of motion by force, a statement about relative motion that would today be called <a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean invariance</a>, and the rule that interactions between bodies do not change the motion of their center of mass. In a later manuscript, Newton added a law of action and reaction, while saying that this law and the law regarding the center of mass implied one another. Newton probably settled on the presentation in the Principia, with three primary laws and then other statements reduced to corollaries, during 1685.<a href="#cite_note-140">[123]</a> <div class="mw-heading mw-heading3"><h3 id="After_the_Principia">After the Principia</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Page_157_from_Mechanism_of_the_Heaven,_Mary_Somerville_1831.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/0/08/Page_157_from_Mechanism_of_the_Heaven%2C_Mary_Somerville_1831.png/220px-Page_157_from_Mechanism_of_the_Heaven%2C_Mary_Somerville_1831.png" decoding="async" width="220" height="367" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/0/08/Page_157_from_Mechanism_of_the_Heaven%2C_Mary_Somerville_1831.png/330px-Page_157_from_Mechanism_of_the_Heaven%2C_Mary_Somerville_1831.png 1.5x, //upload.wikimedia.org/wikipedia/en/0/08/Page_157_from_Mechanism_of_the_Heaven%2C_Mary_Somerville_1831.png 2x" data-file-width="331" data-file-height="552" /></a><figcaption>Page 157 from Mechanism of the Heavens (1831), <a href="/wiki/Mary_Somerville" title="Mary Somerville">Mary Somerville</a>'s expanded version of the first two volumes of Laplace's Traité de mécanique céleste.<a href="#cite_note-141">[124]</a> Here, Somerville deduces the inverse-square law of gravity from <a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a>.</figcaption></figure> Newton expressed his second law by saying that the force on a body is proportional to its change of motion, or momentum. By the time he wrote the Principia, he had already developed calculus (which he called "<a href="/wiki/Fluxion" title="Fluxion">the science of fluxions</a>"), but in the Principia he made no explicit use of it, perhaps because he believed geometrical arguments in the tradition of <a href="/wiki/Euclid" title="Euclid">Euclid</a> to be more rigorous.<a href="#cite_note-142">[125]</a>: 15 <a href="#cite_note-143">[126]</a> Consequently, the Principia does not express acceleration as the second derivative of position, and so it does not give the second law as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=ma}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>m</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=ma}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca4e42b7d6d66f52294364928cb5f7c590f514c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.109ex; height:2.176ex;" alt="{\displaystyle F=ma}">. This form of the second law was written (for the special case of constant force) at least as early as 1716, by <a href="/wiki/Jakob_Hermann" title="Jakob Hermann">Jakob Hermann</a>; <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> would employ it as a basic premise in the 1740s.<a href="#cite_note-144">[127]</a> Euler pioneered the study of rigid bodies<a href="#cite_note-145">[128]</a> and established the basic theory of fluid dynamics.<a href="#cite_note-146">[129]</a> <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a>'s five-volume <a href="/wiki/Trait%C3%A9_de_m%C3%A9canique_c%C3%A9leste" title="Traité de mécanique céleste">Traité de mécanique céleste</a> (1798–1825) forsook geometry and developed mechanics purely through algebraic expressions, while resolving questions that the Principia had left open, like a full theory of the <a href="/wiki/Tide" title="Tide">tides</a>.<a href="#cite_note-147">[130]</a> The concept of energy became a key part of Newtonian mechanics in the post-Newton period. Huygens' solution of the collision of hard spheres showed that in that case, not only is momentum conserved, but kinetic energy is as well (or, rather, a quantity that in retrospect we can identify as one-half the total kinetic energy). The question of what is conserved during all other processes, like inelastic collisions and motion slowed by friction, was not resolved until the 19th century. Debates on this topic overlapped with philosophical disputes between the metaphysical views of Newton and Leibniz, and variants of the term "force" were sometimes used to denote what we would call types of energy. For example, in 1742, <a href="/wiki/%C3%89milie_du_Ch%C3%A2telet" title="Émilie du Châtelet">Émilie du Châtelet</a> wrote, "Dead force consists of a simple tendency to motion: such is that of a spring ready to relax; <a href="/wiki/Vis_viva" title="Vis viva">living force</a> is that which a body has when it is in actual motion." In modern terminology, "dead force" and "living force" correspond to potential energy and kinetic energy respectively.<a href="#cite_note-148">[131]</a> Conservation of energy was not established as a universal principle until it was understood that the energy of mechanical work can be dissipated into heat.<a href="#cite_note-149">[132]</a><a href="#cite_note-150">[133]</a> With the concept of energy given a solid grounding, Newton's laws could then be derived within formulations of classical mechanics that put energy first, as in the Lagrangian and Hamiltonian formulations described above. Modern presentations of Newton's laws use the mathematics of vectors, a topic that was not developed until the late 19th and early 20th centuries. Vector algebra, pioneered by <a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Josiah Willard Gibbs</a> and <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a>, stemmed from and largely supplanted the earlier system of <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> invented by <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a>.<a href="#cite_note-151">[134]</a><a href="#cite_note-152">[135]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><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/History_of_classical_mechanics" title="History of classical mechanics">History of classical mechanics</a></li> <li><a href="/wiki/List_of_eponymous_laws" title="List of eponymous laws">List of eponymous laws</a></li> <li><a href="/wiki/List_of_equations_in_classical_mechanics" title="List of equations in classical mechanics">List of equations in classical mechanics</a></li> <li><a href="/wiki/List_of_scientific_laws_named_after_people" title="List of scientific laws named after people">List of scientific laws named after people</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">List of textbooks on classical mechanics and quantum mechanics</a></li> <li><a href="/wiki/Norton%27s_dome" title="Norton's dome">Norton's dome</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-6"><a href="#cite_ref-6">^</a> See, for example, Zain.<a href="#cite_note-4">[4]</a>: 1-2  <a href="/wiki/David_Tong_(physicist)" title="David Tong (physicist)">David Tong</a> observes, "A particle is defined to be an object of insignificant size: e.g. an electron, a tennis ball or a planet. Obviously the validity of this statement depends on the context..."<a href="#cite_note-5">[5]</a> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Negative acceleration includes both slowing down (when the current velocity is positive) and speeding up (when the current velocity is negative). For this and other points that students have often found difficult, see McDermott et al.<a href="#cite_note-9">[8]</a> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> Per Cohen and Whitman.<a href="#cite_note-Cohen&Whitman-2">[2]</a> For other phrasings, see Eddington<a href="#cite_note-15">[13]</a> and Frautschi et al.<a href="#cite_note-:0-16">[14]</a>: 114  Andrew Motte's 1729 translation rendered Newton's "nisi quatenus" as unless instead of except insofar, which Hoek argues was erroneous.<a href="#cite_note-17">[15]</a><a href="#cite_note-18">[16]</a> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> One textbook observes that a block sliding down an inclined plane is what "some cynics view as the dullest problem in all of physics".<a href="#cite_note-Kleppner-26">[23]</a>: 70  Another quips, "Nobody will ever know how many minds, eager to learn the secrets of the universe, found themselves studying inclined planes and pulleys instead, and decided to switch to some more interesting profession."<a href="#cite_note-:0-16">[14]</a>: 173  </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> For example, José and Saletan (following <a href="/wiki/Ernst_Mach" title="Ernst Mach">Mach</a> and <a href="/wiki/Leonard_Eisenbud" title="Leonard Eisenbud">Eisenbud</a><a href="#cite_note-28">[24]</a>) take the conservation of momentum as a fundamental physical principle and treat <math 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> </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/d0c60fab89e8c3193952047dc565bcf8d233d115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.121ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} =m\mathbf {a} }"> as a definition of "force".<a href="#cite_note-:2-21">[18]</a>: 9  See also Frautschi et al.,<a href="#cite_note-:0-16">[14]</a>: 134  as well as Feynman, Leighton and Sands,<a href="#cite_note-FLS-29">[25]</a>: 12-1  who argue that the second law is incomplete without a specification of a force by another law, like the law of gravity. Kleppner and Kolenkow argue that the second law is incomplete without the third law: an observer who sees one body accelerate without a matching acceleration of some other body to compensate would conclude, not that a force is acting, but that they are not an inertial observer.<a href="#cite_note-Kleppner-26">[23]</a>: 60  Landau and Lifshitz bypass the question by starting with the Lagrangian formalism rather than the Newtonian.<a href="#cite_note-Landau-30">[26]</a> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> See, for instance, Moebs et al.,<a href="#cite_note-32">[27]</a> Gonick and Huffman,<a href="#cite_note-33">[28]</a> Low and Wilson,<a href="#cite_note-34">[29]</a> Stocklmayer et al.,<a href="#cite_note-35">[30]</a> Hellingman,<a href="#cite_note-36">[31]</a> and Hodanbosi.<a href="#cite_note-37">[32]</a> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> See, for example, Frautschi et al.<a href="#cite_note-:0-16">[14]</a>: 356  </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> For the former, see Greiner,<a href="#cite_note-42">[35]</a> or Wachter and Hoeber.<a href="#cite_note-43">[36]</a> For the latter, see Tait<a href="#cite_note-44">[37]</a> and Heaviside.<a href="#cite_note-45">[38]</a> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> Among the many textbook explanations of this are Frautschi et al.<a href="#cite_note-:0-16">[14]</a>: 104  and Boas.<a href="#cite_note-Boas-50">[42]</a>: 287  </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> Among the many textbook treatments of this point are Hand and Finch<a href="#cite_note-hand-finch-54">[45]</a>: 81  and also Kleppner and Kolenkow.<a href="#cite_note-Kleppner-26">[23]</a>: 103  </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> Treatments can be found in, e.g., Chabay et al.<a href="#cite_note-57">[47]</a> and McCallum et al.<a href="#cite_note-58">[48]</a>: 449  </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> Discussions can be found in, for example, Frautschi et al.,<a href="#cite_note-:0-16">[14]</a>: 215  Panofsky and Phillips,<a href="#cite_note-Panofsky1962-88">[77]</a>: 272  Goldstein, Poole and Safko,<a href="#cite_note-:6-90">[79]</a>: 277  and Werner.<a href="#cite_note-91">[80]</a> </li> <li id="cite_note-107"><a href="#cite_ref-107">^</a> Details can be found in the textbooks by, e.g., Cohen-Tannoudji et al.<a href="#cite_note-Cohen-Tannoudji-105">[93]</a>: 242  and Peres.<a href="#cite_note-106">[94]</a>: 302  </li> <li id="cite_note-110"><a href="#cite_ref-110">^</a> As one physicist writes, "Physical theory is possible because we are immersed and included in the whole process – because we can act on objects around us. Our ability to intervene in nature clarifies even the motion of the planets around the sun – masses so great and distances so vast that our roles as participants seem insignificant. Newton was able to transform Kepler's kinematical description of the solar system into a far more powerful dynamical theory because he added concepts from Galileo's experimental methods – force, mass, momentum, and gravitation. The truly external observer will only get as far as Kepler. Dynamical concepts are formulated on the basis of what we can set up, control, and measure."<a href="#cite_note-108">[95]</a> See, for example, Caspar and Hellman.<a href="#cite_note-109">[96]</a> </li> <li id="cite_note-113"><a href="#cite_ref-113">^</a> Aristotelian physics also had difficulty explaining buoyancy, a point that Galileo tried to resolve without complete success.<a href="#cite_note-112">[98]</a> </li> <li id="cite_note-116"><a href="#cite_ref-116">^</a> <a href="/wiki/Anneliese_Maier" title="Anneliese Maier">Anneliese Maier</a> cautions, "Impetus is neither a force, nor a form of energy, nor momentum in the modern sense; it shares something with all these other concepts, but it is identical with none of them."<a href="#cite_note-115">[100]</a>: 79  </li> <li id="cite_note-132"><a href="#cite_ref-132">^</a> Newton himself was an enthusiastic <a href="/wiki/Alchemy" title="Alchemy">alchemist</a>. <a href="/wiki/John_Maynard_Keynes" title="John Maynard Keynes">John Maynard Keynes</a> called him "the last of the magicians" to describe his place in the transition between <a href="/wiki/Protoscience" title="Protoscience">protoscience</a> and modern science.<a href="#cite_note-126">[110]</a><a href="#cite_note-127">[111]</a> The suggestion has been made that alchemy inspired Newton's notion of "action at a distance", i.e., one body exerting a force upon another without being in direct contact.<a href="#cite_note-128">[112]</a> This suggestion enjoyed considerable support among historians of science<a href="#cite_note-129">[113]</a> until a more extensive study of Newton's papers became possible, after which it fell out of favor. However, it does appear that Newton's alchemy influenced his <a href="/wiki/Optics" title="Optics">optics</a>, in particular, how he thought about the combination of colors.<a href="#cite_note-Newman2016-130">[114]</a><a href="#cite_note-131">[115]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Thornton-1"><a href="#cite_ref-Thornton_1-0">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFThorntonMarion2004" class="citation book cs1">Thornton, Stephen T.; Marion, Jerry B. 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Reidel. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-277-0107-5" title="Special:BookSources/90-277-0107-5"><bdi>90-277-0107-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/844001">844001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+principles+of+electromagnetic+theory+and+of+relativity.&rft.place=Dordrecht&rft.pub=D.+Reidel&rft.date=1966&rft_id=info%3Aoclcnum%2F844001&rft.isbn=90-277-0107-5&rft.aulast=Tonnelat&rft.aufirst=Marie-Antoinette&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F844001&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-87"><a href="#cite_ref-87">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChuLebrilla2010" class="citation book cs1">Chu, Caroline S.; Lebrilla, Carlito B. 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Mineola, N.Y.: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-43924-0" title="Special:BookSources/0-486-43924-0"><bdi>0-486-43924-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/56526974">56526974</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Electricity+and+Magnetism&rft.place=Mineola%2C+N.Y.&rft.edition=2nd&rft.pub=Dover+Publications&rft.date=2005&rft_id=info%3Aoclcnum%2F56526974&rft.isbn=0-486-43924-0&rft.aulast=Panofsky&rft.aufirst=Wolfgang+K.+H.&rft.au=Phillips%2C+Melba&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F56526974&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBongaPoissonYang2018" class="citation journal cs1">Bonga, Béatrice; Poisson, Eric; Yang, Huan (November 2018). <a rel="nofollow" class="external text" href="http://aapt.scitation.org/doi/10.1119/1.5054590">"Self-torque and angular momentum balance for a spinning charged sphere"</a>. <a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a>. 86 (11): 839–848. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1805.01372">1805.01372</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/2018AmJPh..86..839B">2018AmJPh..86..839B</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%2F1.5054590">10.1119/1.5054590</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>. <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:53625857">53625857</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=Self-torque+and+angular+momentum+balance+for+a+spinning+charged+sphere&rft.volume=86&rft.issue=11&rft.pages=839-848&rft.date=2018-11&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53625857%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2018AmJPh..86..839B&rft_id=info%3Aarxiv%2F1805.01372&rft.issn=0002-9505&rft_id=info%3Adoi%2F10.1119%2F1.5054590&rft.aulast=Bonga&rft.aufirst=B%C3%A9atrice&rft.au=Poisson%2C+Eric&rft.au=Yang%2C+Huan&rft_id=http%3A%2F%2Faapt.scitation.org%2Fdoi%2F10.1119%2F1.5054590&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-:6-90">^ <a href="#cite_ref-:6_90-0">a</a> <a href="#cite_ref-:6_90-1">b</a> <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>; Poole, Charles P.; Safko, John L. 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San Francisco: Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-31611-0" title="Special:BookSources/0-201-31611-0"><bdi>0-201-31611-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/47056311">47056311</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&rft.edition=3rd&rft.pub=Addison+Wesley&rft.date=2002&rft_id=info%3Aoclcnum%2F47056311&rft.isbn=0-201-31611-0&rft.aulast=Goldstein&rft.aufirst=Herbert&rft.au=Poole%2C+Charles+P.&rft.au=Safko%2C+John+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWerner2014" class="citation journal cs1">Werner, Reinhard F. 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Oxford: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-155226-7" title="Special:BookSources/978-0-19-155226-7"><bdi>978-0-19-155226-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/317496332">317496332</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+and+the+Einstein+Equations&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=2009&rft_id=info%3Aoclcnum%2F317496332&rft.isbn=978-0-19-155226-7&rft.aulast=Choquet-Bruhat&rft.aufirst=Yvonne&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F317496332&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-94"><a href="#cite_ref-94">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEllisWilliams2000" class="citation book cs1"><a href="/wiki/George_F._R._Ellis" title="George F. 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Providence, Rhode Island: American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-6313-7" title="Special:BookSources/978-1-4704-6313-7"><bdi>978-1-4704-6313-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1202475208">1202475208</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Curvature+of+Space+and+Time%2C+with+an+Introduction+to+Geometric+Analysis&rft.place=Providence%2C+Rhode+Island&rft.pub=American+Mathematical+Society&rft.date=2020&rft_id=info%3Aoclcnum%2F1202475208&rft.isbn=978-1-4704-6313-7&rft.aulast=Stavrov&rft.aufirst=Iva&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-Wheeler-98">^ <a href="#cite_ref-Wheeler_98-0">a</a> <a href="#cite_ref-Wheeler_98-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWheeler2010" class="citation book cs1"><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler, John Archibald</a> (18 June 2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zGFkK2tTXPsC&pg=PA235">Geons, Black Holes, and Quantum Foam: A Life in Physics</a>. 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Norton & Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-07948-7" title="Special:BookSources/978-0-393-07948-7"><bdi>978-0-393-07948-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geons%2C+Black+Holes%2C+and+Quantum+Foam%3A+A+Life+in+Physics&rft.pub=W.+W.+Norton+%26+Company&rft.date=2010-06-18&rft.isbn=978-0-393-07948-7&rft.aulast=Wheeler&rft.aufirst=John+Archibald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzGFkK2tTXPsC%26pg%3DPA235&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-99"><a href="#cite_ref-99">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKersting2019" class="citation journal cs1">Kersting, Magdalena (May 2019). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1361-6552%2Fab08f5">"Free fall in curved spacetime—how to visualise gravity in general relativity"</a>. <a href="/wiki/Physics_Education" title="Physics Education">Physics Education</a>. 54 (3): 035008. <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/2019PhyEd..54c5008K">2019PhyEd..54c5008K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1361-6552%2Fab08f5">10.1088/1361-6552/ab08f5</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10852%2F74677">10852/74677</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/0031-9120">0031-9120</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:127471222">127471222</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Education&rft.atitle=Free+fall+in+curved+spacetime%E2%80%94how+to+visualise+gravity+in+general+relativity&rft.volume=54&rft.issue=3&rft.pages=035008&rft.date=2019-05&rft_id=info%3Ahdl%2F10852%2F74677&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A127471222%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2019PhyEd..54c5008K&rft.issn=0031-9120&rft_id=info%3Adoi%2F10.1088%2F1361-6552%2Fab08f5&rft.aulast=Kersting&rft.aufirst=Magdalena&rft_id=https%3A%2F%2Fdoi.org%2F10.1088%252F1361-6552%252Fab08f5&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-100"><a href="#cite_ref-100">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPrescod-Weinstein2021" class="citation book cs1"><a href="/wiki/Chanda_Prescod-Weinstein" title="Chanda Prescod-Weinstein">Prescod-Weinstein, Chanda</a> (2021). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/1164503847">The Disordered Cosmos: A Journey into Dark Matter, Spacetime, and Dreams Deferred</a>. 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John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-16433-X" title="Special:BookSources/0-471-16433-X"><bdi>0-471-16433-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.pub=John+Wiley+%26+Sons&rft.date=2005&rft.isbn=0-471-16433-X&rft.aulast=Cohen-Tannoudji&rft.aufirst=Claude&rft.au=Diu%2C+Bernard&rft.au=Lalo%C3%AB%2C+Franck&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-106"><a href="#cite_ref-106">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeres1993" class="citation book cs1"><a href="/wiki/Asher_Peres" title="Asher Peres">Peres, Asher</a> (1993). <a href="/wiki/Quantum_Theory:_Concepts_and_Methods" title="Quantum Theory: Concepts and Methods">Quantum Theory: Concepts and Methods</a>. <a href="/wiki/Kluwer" class="mw-redirect" title="Kluwer">Kluwer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-2549-4" title="Special:BookSources/0-7923-2549-4"><bdi>0-7923-2549-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/28854083">28854083</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory%3A+Concepts+and+Methods&rft.pub=Kluwer&rft.date=1993&rft_id=info%3Aoclcnum%2F28854083&rft.isbn=0-7923-2549-4&rft.aulast=Peres&rft.aufirst=Asher&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-108"><a href="#cite_ref-108">^</a> D. Bilodeau, quoted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFuchs2011" class="citation book cs1">Fuchs, Christopher A. (6 January 2011). Coming of Age with Quantum Information. Cambridge University Press. pp. 310–311. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-19926-1" title="Special:BookSources/978-0-521-19926-1"><bdi>978-0-521-19926-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/759812415">759812415</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Coming+of+Age+with+Quantum+Information&rft.pages=310-311&rft.pub=Cambridge+University+Press&rft.date=2011-01-06&rft_id=info%3Aoclcnum%2F759812415&rft.isbn=978-0-521-19926-1&rft.aulast=Fuchs&rft.aufirst=Christopher+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-109"><a href="#cite_ref-109">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCaspar2012" class="citation book cs1">Caspar, Max (2012) [1959]. Kepler. Translated by <a href="/wiki/C._Doris_Hellman" title="C. Doris Hellman">Hellman, C. Doris</a>. Dover. p. 178. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-15175-5" title="Special:BookSources/978-0-486-15175-5"><bdi>978-0-486-15175-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/874097920">874097920</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kepler&rft.pages=178&rft.pub=Dover&rft.date=2012&rft_id=info%3Aoclcnum%2F874097920&rft.isbn=978-0-486-15175-5&rft.aulast=Caspar&rft.aufirst=Max&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-111"><a href="#cite_ref-111">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUgaglia2015" class="citation journal cs1">Ugaglia, Monica (2015). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/43915795">"Aristotle's Hydrostatical Physics"</a>. Annali della Scuola Normale Superiore di Pisa. Classe di Lettere e Filosofia. 7 (1): 169–199. <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/0392-095X">0392-095X</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/43915795">43915795</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annali+della+Scuola+Normale+Superiore+di+Pisa.+Classe+di+Lettere+e+Filosofia&rft.atitle=Aristotle%27s+Hydrostatical+Physics&rft.volume=7&rft.issue=1&rft.pages=169-199&rft.date=2015&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F43915795%23id-name%3DJSTOR&rft.issn=0392-095X&rft.aulast=Ugaglia&rft.aufirst=Monica&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F43915795&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-112"><a href="#cite_ref-112">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStraulinoGambiRighini2011" class="citation journal cs1">Straulino, S.; Gambi, C. M. C.; Righini, A. (January 2011). <a rel="nofollow" class="external text" href="http://aapt.scitation.org/doi/10.1119/1.3492721">"Experiments on buoyancy and surface tension following Galileo Galilei"</a>. <a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a>. 79 (1): 32–36. <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/2011AmJPh..79...32S">2011AmJPh..79...32S</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%2F1.3492721">10.1119/1.3492721</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2158%2F530056">2158/530056</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>. <q>Aristotle in his Physics affirmed that solid water should have a greater weight than liquid water for the same volume. We know that this statement is incorrect because the density of ice is lower than that of water (hydrogen bonds create an open crystal structure in the solid phase), and for this reason ice can float. [...] The Aristotelian theory of buoyancy affirms that bodies in a fluid are supported by the resistance of the fluid to being divided by the penetrating object, just as a large piece of wood supports an axe striking it or honey supports a spoon. According to this theory, a boat should sink in shallow water more than in high seas, just as an axe can easily penetrate and even break a small piece of wood, but cannot penetrate a large piece.</q></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=Experiments+on+buoyancy+and+surface+tension+following+Galileo+Galilei&rft.volume=79&rft.issue=1&rft.pages=32-36&rft.date=2011-01&rft_id=info%3Ahdl%2F2158%2F530056&rft.issn=0002-9505&rft_id=info%3Adoi%2F10.1119%2F1.3492721&rft_id=info%3Abibcode%2F2011AmJPh..79...32S&rft.aulast=Straulino&rft.aufirst=S.&rft.au=Gambi%2C+C.+M.+C.&rft.au=Righini%2C+A.&rft_id=http%3A%2F%2Faapt.scitation.org%2Fdoi%2F10.1119%2F1.3492721&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-114"><a href="#cite_ref-114">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSorabji2010" class="citation book cs1">Sorabji, Richard (2010). "John Philoponus". Philoponus and the Rejection of Aristotelian Science (2nd ed.). Institute of Classical Studies, University of London. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-905-67018-5" title="Special:BookSources/978-1-905-67018-5"><bdi>978-1-905-67018-5</bdi></a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/44216227">44216227</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/878730683">878730683</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=John+Philoponus&rft.btitle=Philoponus+and+the+Rejection+of+Aristotelian+Science&rft.edition=2nd&rft.pub=Institute+of+Classical+Studies%2C+University+of+London&rft.date=2010&rft_id=info%3Aoclcnum%2F878730683&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F44216227%23id-name%3DJSTOR&rft.isbn=978-1-905-67018-5&rft.aulast=Sorabji&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-115"><a href="#cite_ref-115">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaier1982" class="citation book cs1"><a href="/wiki/Anneliese_Maier" title="Anneliese Maier">Maier, Anneliese</a> (1982). Sargent, Steven D. (ed.). On the Threshold of Exact Science. University of Pennsylvania Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-812-27831-6" title="Special:BookSources/978-0-812-27831-6"><bdi>978-0-812-27831-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/495305340">495305340</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+the+Threshold+of+Exact+Science&rft.pub=University+of+Pennsylvania+Press&rft.date=1982&rft_id=info%3Aoclcnum%2F495305340&rft.isbn=978-0-812-27831-6&rft.aulast=Maier&rft.aufirst=Anneliese&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-117"><a href="#cite_ref-117">^</a> See, for example: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEatonVavruskaWilloughby2019" class="citation journal cs1">Eaton, Philip; Vavruska, Kinsey; Willoughby, Shannon (25 April 2019). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevPhysEducRes.15.010123">"Exploring the preinstruction and postinstruction non-Newtonian world views as measured by the Force Concept Inventory"</a>. <a href="/wiki/Physical_Review_Physics_Education_Research" class="mw-redirect" title="Physical Review Physics Education Research">Physical Review Physics Education Research</a>. 15 (1): 010123. <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/2019PRPER..15a0123E">2019PRPER..15a0123E</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%2FPhysRevPhysEducRes.15.010123">10.1103/PhysRevPhysEducRes.15.010123</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/2469-9896">2469-9896</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:149482566">149482566</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+Physics+Education+Research&rft.atitle=Exploring+the+preinstruction+and+postinstruction+non-Newtonian+world+views+as+measured+by+the+Force+Concept+Inventory&rft.volume=15&rft.issue=1&rft.pages=010123&rft.date=2019-04-25&rft_id=info%3Adoi%2F10.1103%2FPhysRevPhysEducRes.15.010123&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A149482566%23id-name%3DS2CID&rft.issn=2469-9896&rft_id=info%3Abibcode%2F2019PRPER..15a0123E&rft.aulast=Eaton&rft.aufirst=Philip&rft.au=Vavruska%2C+Kinsey&rft.au=Willoughby%2C+Shannon&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRevPhysEducRes.15.010123&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobertsonGoodhewScherrHeron2021" class="citation journal cs1">Robertson, Amy D.; Goodhew, Lisa M.; <a href="/wiki/Rachel_Scherr" title="Rachel Scherr">Scherr, Rachel E.</a>; Heron, Paula R. L. (March 2021). <a rel="nofollow" class="external text" href="https://aapt.scitation.org/doi/10.1119/10.0003660">"Impetus-Like Reasoning as Continuous with Newtonian Physics"</a>. <a href="/wiki/The_Physics_Teacher" title="The Physics Teacher">The Physics Teacher</a>. 59 (3): 185–188. <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%2F10.0003660">10.1119/10.0003660</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/0031-921X">0031-921X</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:233803836">233803836</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Physics+Teacher&rft.atitle=Impetus-Like+Reasoning+as+Continuous+with+Newtonian+Physics&rft.volume=59&rft.issue=3&rft.pages=185-188&rft.date=2021-03&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A233803836%23id-name%3DS2CID&rft.issn=0031-921X&rft_id=info%3Adoi%2F10.1119%2F10.0003660&rft.aulast=Robertson&rft.aufirst=Amy+D.&rft.au=Goodhew%2C+Lisa+M.&rft.au=Scherr%2C+Rachel+E.&rft.au=Heron%2C+Paula+R.+L.&rft_id=https%3A%2F%2Faapt.scitation.org%2Fdoi%2F10.1119%2F10.0003660&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobertsonGoodhewScherrHeron2021" class="citation journal cs1">Robertson, Amy D.; Goodhew, Lisa M.; <a href="/wiki/Rachel_Scherr" title="Rachel Scherr">Scherr, Rachel E.</a>; Heron, Paula R. L. (30 March 2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevPhysEducRes.17.010121">"University student conceptual resources for understanding forces"</a>. <a href="/wiki/Physical_Review_Physics_Education_Research" class="mw-redirect" title="Physical Review Physics Education Research">Physical Review Physics Education Research</a>. 17 (1): 010121. <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/2021PRPER..17a0121R">2021PRPER..17a0121R</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%2FPhysRevPhysEducRes.17.010121">10.1103/PhysRevPhysEducRes.17.010121</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/2469-9896">2469-9896</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:243143427">243143427</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+Physics+Education+Research&rft.atitle=University+student+conceptual+resources+for+understanding+forces&rft.volume=17&rft.issue=1&rft.pages=010121&rft.date=2021-03-30&rft_id=info%3Adoi%2F10.1103%2FPhysRevPhysEducRes.17.010121&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A243143427%23id-name%3DS2CID&rft.issn=2469-9896&rft_id=info%3Abibcode%2F2021PRPER..17a0121R&rft.aulast=Robertson&rft.aufirst=Amy+D.&rft.au=Goodhew%2C+Lisa+M.&rft.au=Scherr%2C+Rachel+E.&rft.au=Heron%2C+Paula+R.+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRevPhysEducRes.17.010121&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"></li></ul> </li> <li id="cite_note-:8-118">^ <a href="#cite_ref-:8_118-0">a</a> <a href="#cite_ref-:8_118-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlackwell1966" class="citation journal cs1">Blackwell, Richard J. 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Retrieved 6 March 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stanford+Encyclopedia+of+Philosophy&rft.atitle=Newton%27s+Philosophiae+Naturalis+Principia+Mathematica&rft.date=2007-12-20&rft.aulast=Smith&rft.aufirst=George&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fnewton-principia%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-145"><a href="#cite_ref-145">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarquinaMarquinaMarquinaHernández-Gómez2017" class="citation journal cs1">Marquina, J. E.; Marquina, M. L.; Marquina, V.; Hernández-Gómez, J. J. 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Retrieved 7 March 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=History+of+the+Conservation+of+Energy%3A+Booms%2C+Blood%2C+and+Beer+%28Part+3%29&rft.date=2019-08-25&rft_id=https%3A%2F%2Fskullsinthestars.com%2F2019%2F08%2F24%2Fhistory-of-the-conservation-of-energy-booms-blood-and-beer-part-3%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-151"><a href="#cite_ref-151">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilvade_Andrade_Martins2002" class="citation journal cs1">Silva, Cibelle Celestino; de Andrade Martins, Roberto (September 2002). <a rel="nofollow" class="external text" href="http://aapt.scitation.org/doi/10.1119/1.1475326">"Polar and axial vectors versus quaternions"</a>. <a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a>. 70 (9): 958–963. <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/2002AmJPh..70..958S">2002AmJPh..70..958S</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%2F1.1475326">10.1119/1.1475326</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=Polar+and+axial+vectors+versus+quaternions&rft.volume=70&rft.issue=9&rft.pages=958-963&rft.date=2002-09&rft.issn=0002-9505&rft_id=info%3Adoi%2F10.1119%2F1.1475326&rft_id=info%3Abibcode%2F2002AmJPh..70..958S&rft.aulast=Silva&rft.aufirst=Cibelle+Celestino&rft.au=de+Andrade+Martins%2C+Roberto&rft_id=http%3A%2F%2Faapt.scitation.org%2Fdoi%2F10.1119%2F1.1475326&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> <li id="cite_note-152"><a href="#cite_ref-152">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReich1996" class="citation book cs1"><a href="/wiki/Karin_Reich" title="Karin Reich">Reich, Karin</a> (1996). "The Emergence of Vector Calculus in Physics: The Early Decades". In Schubring, Gert (ed.). Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar. Boston Studies in the Philosophy of Science. Vol. 187. Kluwer. pp. 197–210. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9-048-14758-8" title="Special:BookSources/978-9-048-14758-8"><bdi>978-9-048-14758-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/799299609">799299609</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Emergence+of+Vector+Calculus+in+Physics%3A+The+Early+Decades&rft.btitle=Hermann+G%C3%BCnther+Gra%C3%9Fmann+%281809%E2%80%931877%29%3A+Visionary+Mathematician%2C+Scientist+and+Neohumanist+Scholar&rft.series=Boston+Studies+in+the+Philosophy+of+Science&rft.pages=197-210&rft.pub=Kluwer&rft.date=1996&rft_id=info%3Aoclcnum%2F799299609&rft.isbn=978-9-048-14758-8&rft.aulast=Reich&rft.aufirst=Karin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <ul><li><a rel="nofollow" class="external text" href="https://www.feynmanlectures.caltech.edu/I_09.html">Newton’s Laws of Dynamics - The Feynman Lectures on Physics</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChakrabartyDourmashkinTomasikFrebel2016" class="citation web cs1">Chakrabarty, Deepto; Dourmashkin, Peter; Tomasik, Michelle; <a href="/wiki/Anna_Frebel" title="Anna Frebel">Frebel, Anna</a>; Vuletic, Vladan (2016). <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/physics/8-01sc-classical-mechanics-fall-2016/index.htm">"Classical Mechanics"</a>. <a href="/wiki/MIT_OpenCourseWare" title="MIT OpenCourseWare">MIT OpenCourseWare</a>. Retrieved 17 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MIT+OpenCourseWare&rft.atitle=Classical+Mechanics&rft.date=2016&rft.aulast=Chakrabarty&rft.aufirst=Deepto&rft.au=Dourmashkin%2C+Peter&rft.au=Tomasik%2C+Michelle&rft.au=Frebel%2C+Anna&rft.au=Vuletic%2C+Vladan&rft_id=https%3A%2F%2Focw.mit.edu%2Fcourses%2Fphysics%2F8-01sc-classical-mechanics-fall-2016%2Findex.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+laws+of+motion" class="Z3988"></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 .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style><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><div role="navigation" class="navbox" aria-labelledby="Sir_Isaac_Newton" 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:Isaac_Newton" title="Template:Isaac Newton"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Isaac_Newton" title="Template talk:Isaac Newton"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Isaac_Newton" title="Special:EditPage/Template:Isaac Newton"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Sir_Isaac_Newton" style="font-size:114%;margin:0 4em"><a href="/wiki/Isaac_Newton" title="Isaac Newton">Sir Isaac Newton</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Publications</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/Method_of_Fluxions" title="Method of Fluxions">Fluxions</a> (1671)</li> <li><a href="/wiki/De_motu_corporum_in_gyrum" title="De motu corporum in gyrum">De Motu</a> (1684)</li> <li><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a> (1687)</li> <li><a href="/wiki/Opticks" title="Opticks">Opticks</a> (1704)</li> <li><a href="/wiki/The_Queries" class="mw-redirect" title="The Queries">Queries</a> (1704)</li> <li><a href="/wiki/Arithmetica_Universalis" title="Arithmetica Universalis">Arithmetica</a> (1707)</li> <li><a href="/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas" title="De analysi per aequationes numero terminorum infinitas">De Analysi</a> (1711)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Other writings</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/Quaestiones_quaedam_philosophicae" title="Quaestiones quaedam philosophicae">Quaestiones</a> (1661–1665)</li> <li>"<a href="/wiki/Standing_on_the_shoulders_of_giants" title="Standing on the shoulders of giants">standing on the shoulders of giants</a>" (1675)</li> <li><a href="/wiki/Notes_on_the_Jewish_Temple" title="Notes on the Jewish Temple">Notes on the Jewish Temple</a> (c. 1680)</li> <li>"<a href="/wiki/General_Scholium" title="General Scholium">General Scholium</a>" (1713; "<a href="/wiki/Hypotheses_non_fingo" title="Hypotheses non fingo">hypotheses non fingo</a>" )</li> <li><a href="/wiki/The_Chronology_of_Ancient_Kingdoms_Amended" title="The Chronology of Ancient Kingdoms Amended">Ancient Kingdoms Amended</a> (1728)</li> <li><a href="/wiki/An_Historical_Account_of_Two_Notable_Corruptions_of_Scripture" title="An Historical Account of Two Notable Corruptions of Scripture">Corruptions of Scripture</a> (1754)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Contributions</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/Calculus" title="Calculus">Calculus</a> <ul><li><a href="/wiki/Fluxion" title="Fluxion">fluxion</a></li></ul></li> <li><a href="/wiki/Impact_depth" title="Impact depth">Impact depth</a></li> <li><a href="/wiki/Inertia" title="Inertia">Inertia</a></li> <li><a href="/wiki/Newton_disc" title="Newton disc">Newton disc</a></li> <li><a href="/wiki/Newton_polygon" title="Newton polygon">Newton polygon</a> <ul><li><a href="/wiki/Newton%E2%80%93Okounkov_body" title="Newton–Okounkov body">Newton–Okounkov body</a></li></ul></li> <li><a href="/wiki/Newton%27s_reflector" title="Newton's reflector">Newton's reflector</a></li> <li><a href="/wiki/Newtonian_telescope" title="Newtonian telescope">Newtonian telescope</a></li> <li><a href="/wiki/Newton_scale" title="Newton scale">Newton scale</a></li> <li><a href="/wiki/Newton%27s_metal" title="Newton's metal">Newton's metal</a></li> <li><a href="/wiki/Spectrum" title="Spectrum">Spectrum</a></li> <li><a href="/wiki/Structural_coloration" title="Structural coloration">Structural coloration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Newtonianism" title="Newtonianism">Newtonianism</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/Bucket_argument" title="Bucket argument">Bucket argument</a></li> <li><a href="/wiki/Newton%27s_inequalities" title="Newton's inequalities">Newton's inequalities</a></li> <li><a href="/wiki/Newton%27s_law_of_cooling" title="Newton's law of cooling">Newton's law of cooling</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> <ul><li><a href="/wiki/Post-Newtonian_expansion" title="Post-Newtonian expansion">post-Newtonian expansion</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">parameterized</a></li> <li><a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a></li></ul></li> <li><a href="/wiki/Newton%E2%80%93Cartan_theory" title="Newton–Cartan theory">Newton–Cartan theory</a></li> <li><a href="/wiki/Schr%C3%B6dinger%E2%80%93Newton_equation" title="Schrödinger–Newton equation">Schrödinger–Newton equation</a></li> <li><a class="mw-selflink selflink">Newton's laws of motion</a> <ul><li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws</a></li></ul></li> <li><a href="/wiki/Newtonian_dynamics" title="Newtonian dynamics">Newtonian dynamics</a></li> <li><a href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method in optimization</a> <ul><li><a href="/wiki/Problem_of_Apollonius" title="Problem of Apollonius">Apollonius's problem</a></li> <li><a href="/wiki/Truncated_Newton_method" title="Truncated Newton method">truncated Newton method</a></li></ul></li> <li><a href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton algorithm</a></li> <li><a href="/wiki/Newton%27s_rings" title="Newton's rings">Newton's rings</a></li> <li><a href="/wiki/Newton%27s_theorem_about_ovals" title="Newton's theorem about ovals">Newton's theorem about ovals</a></li> <li><a href="/wiki/Newton%E2%80%93Pepys_problem" title="Newton–Pepys problem">Newton–Pepys problem</a></li> <li><a href="/wiki/Newtonian_potential" title="Newtonian potential">Newtonian potential</a></li> <li><a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</a></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light">Corpuscular theory of light</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton calculus controversy</a></li> <li><a href="/wiki/Newton%27s_notation" class="mw-redirect" title="Newton's notation">Newton's notation</a></li> <li><a href="/wiki/Rotating_spheres" title="Rotating spheres">Rotating spheres</a></li> <li><a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a></li> <li><a href="/wiki/Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes formulas</a></li> <li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> <ul><li><a href="/wiki/Generalized_Gauss%E2%80%93Newton_method" title="Generalized Gauss–Newton method">generalized Gauss–Newton method</a></li></ul></li> <li><a href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</a></li> <li><a href="/wiki/Newton%27s_identities" title="Newton's identities">Newton's identities</a></li> <li><a href="/wiki/Newton_polynomial" title="Newton polynomial">Newton polynomial</a></li> <li><a href="/wiki/Newton%27s_theorem_of_revolving_orbits" title="Newton's theorem of revolving orbits">Newton's theorem of revolving orbits</a></li> <li><a href="/wiki/Newton%E2%80%93Euler_equations" title="Newton–Euler equations">Newton–Euler equations</a></li> <li><a href="/wiki/Power_number" title="Power number">Newton number</a> <ul><li><a href="/wiki/Kissing_number" title="Kissing number">kissing number problem</a></li></ul></li> <li><a href="/wiki/Difference_quotient" title="Difference quotient">Newton's quotient</a></li> <li><a href="/wiki/Parallelogram_of_force" title="Parallelogram of force">Parallelogram of force</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Newton–Puiseux theorem</a></li> <li><a href="/wiki/Absolute_space_and_time#Newton" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Luminiferous_aether" title="Luminiferous aether">Luminiferous aether</a></li> <li><a href="/wiki/Finite_difference" title="Finite difference">Newtonian series</a> <ul><li><a href="/wiki/Table_of_Newtonian_series" title="Table of Newtonian series">table</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Personal life</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/Woolsthorpe_Manor" title="Woolsthorpe Manor">Woolsthorpe Manor</a> (birthplace)</li> <li><a href="/wiki/Cranbury_Park" title="Cranbury Park">Cranbury Park</a> (home)</li> <li><a href="/wiki/Early_life_of_Isaac_Newton" title="Early life of Isaac Newton">Early life</a></li> <li><a href="/wiki/Later_life_of_Isaac_Newton" title="Later life of Isaac Newton">Later life</a></li> <li><a href="/wiki/Isaac_Newton%27s_apple_tree" title="Isaac Newton's apple tree">Apple tree</a></li> <li><a href="/wiki/Religious_views_of_Isaac_Newton" title="Religious views of Isaac Newton">Religious views</a></li> <li><a href="/wiki/Isaac_Newton%27s_occult_studies" title="Isaac Newton's occult studies">Occult studies</a></li> <li><a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a></li> <li><a href="/wiki/Copernican_Revolution" title="Copernican Revolution">Copernican Revolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Relations</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/Catherine_Barton" title="Catherine Barton">Catherine Barton</a> (niece)</li> <li><a href="/wiki/John_Conduitt" title="John Conduitt">John Conduitt</a> (nephew-in-law)</li> <li><a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Isaac Barrow</a> (professor)</li> <li><a href="/wiki/William_Clarke_(apothecary)" title="William Clarke (apothecary)">William Clarke</a> (mentor)</li> <li><a href="/wiki/Benjamin_Pulleyn" title="Benjamin Pulleyn">Benjamin Pulleyn</a> (tutor)</li> <li><a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a> (student)</li> <li><a href="/wiki/William_Whiston" title="William Whiston">William Whiston</a> (student)</li> <li><a href="/wiki/John_Keill" title="John Keill">John Keill</a> (disciple)</li> <li><a href="/wiki/William_Stukeley" title="William Stukeley">William Stukeley</a> (friend)</li> <li><a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a> (friend)</li> <li><a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> (friend)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Isaac_Newton_in_popular_culture" title="Isaac Newton in popular culture">Depictions</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/Newton_(Blake)" title="Newton (Blake)">Newton by Blake</a> (monotype)</li> <li><a href="/wiki/Newton_(Paolozzi)" title="Newton (Paolozzi)">Newton by Paolozzi</a> (sculpture)</li> <li><a href="/wiki/Isaac_Newton_Gargoyle" title="Isaac Newton Gargoyle">Isaac Newton Gargoyle</a></li> <li><a href="/wiki/Astronomers_Monument" title="Astronomers Monument">Astronomers Monument</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/List_of_things_named_after_Isaac_Newton" title="List of things named after Isaac Newton">Namesake</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/Newton_(unit)" title="Newton (unit)">Newton (unit)</a></li> <li><a href="/wiki/Newton%27s_cradle" title="Newton's cradle">Newton's cradle</a></li> <li><a href="/wiki/Isaac_Newton_Institute" title="Isaac Newton Institute">Isaac Newton Institute</a></li> <li><a href="/wiki/Institute_of_Physics_Isaac_Newton_Medal" class="mw-redirect" title="Institute of Physics Isaac Newton Medal">Isaac Newton Medal</a></li> <li><a href="/wiki/Isaac_Newton_Telescope" title="Isaac Newton Telescope">Isaac Newton Telescope</a></li> <li><a href="/wiki/Isaac_Newton_Group_of_Telescopes" title="Isaac Newton Group of Telescopes">Isaac Newton Group of Telescopes</a></li> <li><a href="/wiki/XMM-Newton" title="XMM-Newton">XMM-Newton</a></li> <li><a href="/wiki/Sir_Isaac_Newton_Sixth_Form" title="Sir Isaac Newton Sixth Form">Sir Isaac Newton Sixth Form</a></li> <li><a href="/wiki/Statal_Institute_of_Higher_Education_Isaac_Newton" title="Statal Institute of Higher Education Isaac Newton">Statal Institute of Higher Education Isaac Newton</a></li> <li><a href="/wiki/Newton_International_Fellowship" title="Newton International Fellowship">Newton International Fellowship</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" 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