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Available in 11 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-11" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">11 languages</span> </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/Euler_se_bewegingswette" title="Euler se bewegingswette – Afrikaans" lang="af" hreflang="af" data-title="Euler se bewegingswette" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86%D9%8A_%D8%A3%D9%88%D9%8A%D9%84%D8%B1_%D9%84%D9%84%D8%AD%D8%B1%D9%83%D8%A9" title="قانوني أويلر للحركة – Arabic" lang="ar" hreflang="ar" data-title="قانوني أويلر للحركة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Lleis_del_moviment_d%27Euler" title="Lleis del moviment d'Euler – Catalan" lang="ca" hreflang="ca" data-title="Lleis del moviment d'Euler" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Cauchy-eulersche_Bewegungsgesetze" title="Cauchy-eulersche Bewegungsgesetze – German" lang="de" hreflang="de" data-title="Cauchy-eulersche Bewegungsgesetze" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></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_%D8%A7%D9%88%DB%8C%D9%84%D8%B1" title="قوانین حرکت اویلر – Persian" lang="fa" hreflang="fa" data-title="قوانین حرکت اویلر" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hukum_gerak_Euler" title="Hukum gerak Euler – Indonesian" lang="id" hreflang="id" data-title="Hukum gerak Euler" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazioni_di_Eulero_(dinamica)" title="Equazioni di Eulero (dinamica) – Italian" lang="it" hreflang="it" data-title="Equazioni di Eulero (dinamica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8_%D0%B7%D0%B0_%D0%B4%D0%B2%D0%B8%D0%B6%D0%B5%D1%9A%D0%B5" title="Ојлерови закони за движење – Macedonian" lang="mk" hreflang="mk" data-title="Ојлерови закони за движење" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Legile_mi%C8%99c%C4%83rii_ale_lui_Euler" title="Legile mișcării ale lui Euler – Romanian" lang="ro" hreflang="ro" data-title="Legile mișcării ale lui Euler" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_lu%E1%BA%ADt_chuy%E1%BB%83n_%C4%91%E1%BB%99ng_c%E1%BB%A7a_Euler" title="Định luật chuyển động của Euler – Vietnamese" lang="vi" hreflang="vi" data-title="Định luật chuyển động của Euler" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AD%90%E6%8B%89%E9%81%8B%E5%8B%95%E5%AE%9A%E5%BE%8B" title="歐拉運動定律 – Chinese" lang="zh" hreflang="zh" data-title="歐拉運動定律" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Extend Newton's laws of motion to rigid bodies</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Euler's first law" and "Euler's second law" redirect here. 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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"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}}"></span><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><span class="nowrap"><a href="/wiki/Inertia" title="Inertia">Inertia</a> / <a href="/wiki/Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></span></li> <li><a href="/wiki/Mass" title="Mass">Mass</a></li> <li><br /><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><br /><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;"><b><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></b></div></li> <li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></b> <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 class="mw-selflink selflink"><span class="wrap">Euler's laws of motion</span></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><span class="nowrap"><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></span></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"><span class="wrap">Newton's law of universal gravitation</span></a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li> <li><a href="/wiki/Relative_velocity" title="Relative velocity">Relative velocity</a></li> <li><a href="/wiki/Rigid_body" title="Rigid body">Rigid body</a> <ul><li><a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li> <li><a href="/wiki/Euler%27s_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li> <li><a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li> <li><a href="/wiki/Vibration" title="Vibration">Vibration</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)"><a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Circular_motion" title="Circular motion">Circular motion</a></li> <li><a href="/wiki/Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li> <li><a href="/wiki/Centripetal_force" title="Centripetal force">Centripetal force</a></li> <li><a href="/wiki/Centrifugal_force" title="Centrifugal force">Centrifugal force</a> <ul><li><a href="/wiki/Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li> <li><a href="/wiki/Tangential_speed" title="Tangential speed">Tangential speed</a></li> <li><a href="/wiki/Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul> </div> <ul><li><a href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="/wiki/Angular_displacement" title="Angular displacement">displacement</a> / <a href="/wiki/Angular_frequency" title="Angular frequency">frequency</a> / <a href="/wiki/Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a></li> <li><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li> <li><a href="/wiki/Edmond_Halley" title="Edmond Halley">Halley</a></li> <li><a href="/wiki/Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li> <li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li> <li><a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">d'Alembert</a></li> <li><a href="/wiki/Alexis_Clairaut" title="Alexis Clairaut">Clairaut</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li> <li><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a></li> <li><a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Edward_Routh" title="Edward Routh">Routh</a></li> <li><a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Liouville</a></li> <li><a href="/wiki/Paul_%C3%89mile_Appell" title="Paul Émile Appell">Appell</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Bernard_Koopman" title="Bernard Koopman">Koopman</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <ul><li><span class="nowrap"><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/14px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="14" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/21px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/28px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span> </span><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Classical_mechanics" title="Category:Classical mechanics">Category</a></span></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classical_mechanics" title="Template:Classical mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classical_mechanics" title="Template talk:Classical mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics" title="Special:EditPage/Template:Classical mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, <b>Euler's laws of motion</b> are <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a> which extend <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> for <a href="/wiki/Point_particle" title="Point particle">point particle</a> to <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a> motion.<sup id="cite_ref-McGillKing_1-0" class="reference"><a href="#cite_note-McGillKing-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> They were formulated by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> about 50 years after <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> formulated his laws. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler%27s_laws_of_motion&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Euler's_first_law"><span id="Euler.27s_first_law"></span>Euler's first law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler%27s_laws_of_motion&action=edit&section=2" title="Edit section: Euler's first law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Euler's first law</b> states that the rate of change of <a href="/wiki/Momentum#Linear_momentum_of_a_system_of_particles" title="Momentum">linear momentum</a> <span class="texhtml"><b>p</b></span> of a rigid body is equal to the resultant of all the external forces <span class="texhtml"><b>F</b><sub>ext</sub></span> acting on the body:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><br /> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\text{ext}}={\frac {d\mathbf {p} }{dt}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ext</mtext> </mrow> </msub> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\text{ext}}={\frac {d\mathbf {p} }{dt}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f91b8a321cc9197f8f64ba4ed714b14818636854" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.247ex; height:5.509ex;" alt="{\displaystyle F_{\text{ext}}={\frac {d\mathbf {p} }{dt}}.}"></span></dd></dl> <p>Internal forces between the particles that make up a body do not contribute to changing the momentum of the body as there is an equal and opposite force resulting in no net effect.<sup id="cite_ref-Gray_3-0" class="reference"><a href="#cite_note-Gray-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The linear momentum of a rigid body is the product of the mass of the body and the velocity of its <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> <span class="texhtml"><b>v</b><sub>cm</sub></span>.<sup id="cite_ref-McGillKing_1-1" class="reference"><a href="#cite_note-McGillKing-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-BookRags_4-0" class="reference"><a href="#cite_note-BookRags-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Rao_5-0" class="reference"><a href="#cite_note-Rao-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Euler's_second_law"><span id="Euler.27s_second_law"></span>Euler's second law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler%27s_laws_of_motion&action=edit&section=3" title="Edit section: Euler's second law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Balance_of_angular_momentum" title="Balance of angular momentum">Balance of angular momentum</a></div> <p><b>Euler's second law</b> states that the rate of change of <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> <span class="texhtml"><b>L</b></span> about a point that is fixed in an inertial reference frame (often the center of mass of the body), is equal to the sum of the external moments of force (<a href="/wiki/Torques" class="mw-redirect" title="Torques">torques</a>) acting on that body <span class="texhtml"><b>M</b></span> about that point:<sup id="cite_ref-McGillKing_1-2" class="reference"><a href="#cite_note-McGillKing-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-BookRags_4-1" class="reference"><a href="#cite_note-BookRags-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Rao_5-1" class="reference"><a href="#cite_note-Rao-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} ={d\mathbf {L} \over dt}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>=</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} ={d\mathbf {L} \over dt}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ab84527d827cc8eb1ae231e3c4f06f422eaef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.943ex; height:5.509ex;" alt="{\displaystyle \mathbf {M} ={d\mathbf {L} \over dt}.}"></span></dd></dl> <p>Note that the above formula holds only if both <span class="texhtml"><b>M</b></span> and <span class="texhtml"><b>L</b></span> are computed with respect to a fixed inertial frame or a frame parallel to the inertial frame but fixed on the center of mass. For rigid bodies translating and rotating in only two dimensions, this can be expressed as:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} =\mathbf {r} _{\rm {cm}}\times \mathbf {a} _{\rm {cm}}m+I{\boldsymbol {\alpha }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>m</mi> <mo>+</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} =\mathbf {r} _{\rm {cm}}\times \mathbf {a} _{\rm {cm}}m+I{\boldsymbol {\alpha }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46425d50afd8692cf756adffed1e493f4429cbee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} =\mathbf {r} _{\rm {cm}}\times \mathbf {a} _{\rm {cm}}m+I{\boldsymbol {\alpha }},}"></span></dd></dl> <p>where: </p> <ul><li><span class="texhtml"><b>r</b><sub>cm</sub></span> is the position vector of the center of mass of the body with respect to the point about which moments are summed,</li> <li><span class="texhtml"><b>a</b><sub>cm</sub></span> is the linear acceleration of the center of mass of the body,</li> <li><span class="texhtml"><b>m</b></span> is the mass of the body,</li> <li><span class="texhtml"><b>α</b></span> is the <a href="/wiki/Angular_acceleration" title="Angular acceleration">angular acceleration</a> of the body, and</li> <li><span class="texhtml"><b>I</b></span> is the <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> of the body about its center of mass.</li></ul> <p>See also <a href="/wiki/Euler%27s_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations (rigid body dynamics)</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Explanation_and_derivation">Explanation and derivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler%27s_laws_of_motion&action=edit&section=4" title="Edit section: Explanation and derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The distribution of internal forces in a deformable body are not necessarily equal throughout, i.e. the stresses vary from one point to the next. This variation of internal forces throughout the body is governed by <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's second law of motion</a> of conservation of <a href="/wiki/Linear_momentum" class="mw-redirect" title="Linear momentum">linear momentum</a> and <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>, which for their simplest use are applied to a mass particle but are extended in <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a> to a body of continuously distributed mass. For continuous bodies these laws are called <b>Euler's laws of motion</b>.<sup id="cite_ref-Lubliner_7-0" class="reference"><a href="#cite_note-Lubliner-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The total body force applied to a continuous body with mass <span class="texhtml"><i>m</i></span>, <a href="/wiki/Mass_density" class="mw-redirect" title="Mass density">mass density</a> <span class="texhtml"><i>ρ</i></span>, and volume <span class="texhtml"><i>V</i></span>, is the <a href="/wiki/Volume_integral" title="Volume integral">volume integral</a> integrated over the volume of the body: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{B}=\int _{V}\mathbf {b} \,dm=\int _{V}\mathbf {b} \rho \,dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>m</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{B}=\int _{V}\mathbf {b} \,dm=\int _{V}\mathbf {b} \rho \,dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c794458953062f48e78af3a901df549ba096189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.917ex; height:5.676ex;" alt="{\displaystyle \mathbf {F} _{B}=\int _{V}\mathbf {b} \,dm=\int _{V}\mathbf {b} \rho \,dV}"></span></dd></dl> <p>where <span class="texhtml"><b>b</b></span> is the force acting on the body per unit mass (<a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensions</a> of acceleration, misleadingly called the "body force"), and <span class="texhtml"><i>dm</i> = <i>ρ dV</i></span> is an infinitesimal mass element of the body. </p><p>Body forces and contact forces acting on the body lead to corresponding moments (<a href="/wiki/Torque" title="Torque">torques</a>) of those forces relative to a given point. Thus, the total applied torque <span class="texhtml"><b>M</b></span> about the origin is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8790179d7bcfb6b691fb7e0aa51e11da5715c241" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.512ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}}"></span></dd></dl> <p>where <span class="texhtml"><b>M</b><sub><i>B</i></sub></span> and <span class="texhtml"><b>M</b><sub><i>C</i></sub></span> respectively indicate the moments caused by the body and contact forces. </p><p>Thus, the sum of all applied forces and torques (with respect to the origin of the coordinate system) acting on the body can be given as the sum of a volume and <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\int _{V}\mathbf {a} \,dm=\int _{V}\mathbf {a} \rho \,dV=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>m</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>S</mi> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\int _{V}\mathbf {a} \,dm=\int _{V}\mathbf {a} \rho \,dV=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651922868b07be2854bf7cf5ce6308962c7a64cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.371ex; height:5.676ex;" alt="{\displaystyle \mathbf {F} =\int _{V}\mathbf {a} \,dm=\int _{V}\mathbf {a} \rho \,dV=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>S</mi> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20c85666ade3d360798ffd042604fb2e11b93664" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.349ex; height:5.676ex;" alt="{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV.}"></span></dd></dl> <p>where <span class="texhtml"><b>t</b> = <b>t</b>(<b>n</b>)</span> is called the <a href="/wiki/Surface_traction" class="mw-redirect" title="Surface traction">surface traction</a>, integrated over the surface of the body, in turn <span class="texhtml"><b>n</b></span> denotes a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> normal and directed outwards to the surface <span class="texhtml"><i>S</i></span>. </p><p>Let the coordinate system <span class="texhtml">(<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>)</span> be an <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial frame of reference</a>, <span class="texhtml"><b>r</b></span> be the position vector of a point particle in the continuous body with respect to the origin of the coordinate system, and <span class="texhtml"><b>v</b> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>d</i><b>r</b></span><span class="sr-only">/</span><span class="den"><i>dt</i></span></span>⁠</span></span> be the velocity vector of that point. </p><p><b>Euler's first axiom or law</b> (law of balance of linear momentum or balance of forces) states that in an inertial frame the time rate of change of linear momentum <span class="texhtml"><b>p</b></span> of an arbitrary portion of a continuous body is equal to the total applied force <span class="texhtml"><b>F</b></span> acting on that portion, and it is expressed as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d\mathbf {p} }{dt}}&=\mathbf {F} \\{\frac {d}{dt}}\int _{V}\rho \mathbf {v} \,dV&=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>S</mi> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d\mathbf {p} }{dt}}&=\mathbf {F} \\{\frac {d}{dt}}\int _{V}\rho \mathbf {v} \,dV&=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b11d4cc87bdbec4fa9604c4abd74741505c9ba24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.902ex; margin-bottom: -0.269ex; width:36.161ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {d\mathbf {p} }{dt}}&=\mathbf {F} \\{\frac {d}{dt}}\int _{V}\rho \mathbf {v} \,dV&=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV.\end{aligned}}}"></span></dd></dl> <p><b>Euler's second axiom or law</b> (law of balance of angular momentum or balance of torques) states that in an inertial frame the time rate of change of angular momentum <span class="texhtml"><b>L</b></span> of an arbitrary portion of a continuous body is equal to the total applied torque <span class="texhtml"><b>M</b></span> acting on that portion, and it is expressed as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d\mathbf {L} }{dt}}&=\mathbf {M} \\{\frac {d}{dt}}\int _{V}\mathbf {r} \times \rho \mathbf {v} \,dV&=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>S</mi> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d\mathbf {L} }{dt}}&=\mathbf {M} \\{\frac {d}{dt}}\int _{V}\mathbf {r} \times \rho \mathbf {v} \,dV&=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a636166f28698822de80eacf89d9f021710838b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:47.989ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {d\mathbf {L} }{dt}}&=\mathbf {M} \\{\frac {d}{dt}}\int _{V}\mathbf {r} \times \rho \mathbf {v} \,dV&=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV.\end{aligned}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> is the velocity, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> the volume, and the derivatives of <span class="texhtml"><b>p</b></span> and <span class="texhtml"><b>L</b></span> are <a href="/wiki/Material_derivative" title="Material derivative">material derivatives</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler%27s_laws_of_motion&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_topics_named_after_Leonhard_Euler" title="List of topics named after Leonhard Euler">List of topics named after Leonhard Euler</a></li> <li><a href="/wiki/Euler%27s_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's laws of rigid body rotations</a></li> <li><a href="/wiki/Newton%E2%80%93Euler_equations" title="Newton–Euler equations">Newton–Euler equations</a> of motion with 6 components, combining Euler's two laws into one equation.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler%27s_laws_of_motion&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-McGillKing-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-McGillKing_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-McGillKing_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-McGillKing_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><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="CITEREFMcGill_and_King1995" class="citation book cs1">McGill and King (1995). <i>Engineering Mechanics, An Introduction to Dynamics</i> (3rd ed.). PWS Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-93399-8" title="Special:BookSources/0-534-93399-8"><bdi>0-534-93399-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics%2C+An+Introduction+to+Dynamics&rft.edition=3rd&rft.pub=PWS+Publishing+Company&rft.date=1995&rft.isbn=0-534-93399-8&rft.au=McGill+and+King&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+laws+of+motion" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://emweb.unl.edu/NEGAHBAN/EM373/note19/note19.htm">Equations of motion for a rigid body</a> Retrieved 2021-06-06</span> </li> <li id="cite_note-Gray-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gray_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrayCostanzo,_Plesha2010" class="citation book cs1">Gray, Gary L.; Costanzo, Plesha (2010). <i>Engineering Mechanics: Dynamics</i>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-282871-9" title="Special:BookSources/978-0-07-282871-9"><bdi>978-0-07-282871-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics%3A+Dynamics&rft.pub=McGraw-Hill&rft.date=2010&rft.isbn=978-0-07-282871-9&rft.aulast=Gray&rft.aufirst=Gary+L.&rft.au=Costanzo%2C+Plesha&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+laws+of+motion" class="Z3988"></span></span> </li> <li id="cite_note-BookRags-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-BookRags_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BookRags_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://www.bookrags.com/research/eulers-laws-of-motion-wom/"><i>Euler's Laws of Motion</i></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2009-03-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euler%27s+Laws+of+Motion&rft_id=http%3A%2F%2Fwww.bookrags.com%2Fresearch%2Feulers-laws-of-motion-wom%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+laws+of+motion" class="Z3988"></span></span> </li> <li id="cite_note-Rao-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rao_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rao_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRao2006" class="citation book cs1">Rao, Anil Vithala (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2y9e6BjxZf4C&q=euler's+laws&pg=PA355"><i>Dynamics of particles and rigid bodies</i></a>. Cambridge University Press. p. 355. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-85811-3" title="Special:BookSources/978-0-521-85811-3"><bdi>978-0-521-85811-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics+of+particles+and+rigid+bodies&rft.pages=355&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-85811-3&rft.aulast=Rao&rft.aufirst=Anil+Vithala&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2y9e6BjxZf4C%26q%3Deuler%27s%2Blaws%26pg%3DPA355&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+laws+of+motion" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRuinaRudra_Pratap2002" class="citation book cs1">Ruina, Andy; <a href="/wiki/Rudra_Pratap" title="Rudra Pratap">Rudra Pratap</a> (2002). <a rel="nofollow" class="external text" href="http://ruina.tam.cornell.edu/Book/RuinaPratapNoProblems.pdf"><i>Introduction to Statics and Dynamics</i></a> <span class="cs1-format">(PDF)</span>. Oxford University Press. p. 771<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-10-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Statics+and+Dynamics&rft.pages=771&rft.pub=Oxford+University+Press&rft.date=2002&rft.aulast=Ruina&rft.aufirst=Andy&rft.au=Rudra+Pratap&rft_id=http%3A%2F%2Fruina.tam.cornell.edu%2FBook%2FRuinaPratapNoProblems.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler%27s+laws+of+motion" class="Z3988"></span></span> </li> <li id="cite_note-Lubliner-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lubliner_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLubliner2008" class="citation book cs1">Lubliner, Jacob (2008). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100331022415/http://www.ce.berkeley.edu/~coby/plas/pdf/book.pdf"><i>Plasticity Theory</i></a> <span class="cs1-format">(PDF)</span> (Revised ed.). Dover Publications. pp. 27–28. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-46290-5" title="Special:BookSources/978-0-486-46290-5"><bdi>978-0-486-46290-5</bdi></a>. 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