Rigid body dynamics

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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Planar_rigid_body_dynamics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Planar_rigid_body_dynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Planar rigid body dynamics</span> </div> </a> <ul id="toc-Planar_rigid_body_dynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rigid_body_in_three_dimensions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Rigid_body_in_three_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Rigid body in three dimensions</span> </div> </a> <button aria-controls="toc-Rigid_body_in_three_dimensions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Rigid body in three dimensions subsection</span> </button> <ul id="toc-Rigid_body_in_three_dimensions-sublist" class="vector-toc-list"> <li id="toc-Orientation_or_attitude_descriptions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orientation_or_attitude_descriptions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Orientation or attitude descriptions</span> </div> </a> <ul id="toc-Orientation_or_attitude_descriptions-sublist" class="vector-toc-list"> <li id="toc-Euler_angles" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Euler_angles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Euler angles</span> </div> </a> <ul id="toc-Euler_angles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tait–Bryan_angles" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Tait–Bryan_angles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Tait–Bryan angles</span> </div> </a> <ul id="toc-Tait–Bryan_angles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orientation_vector" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Orientation_vector"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Orientation vector</span> </div> </a> <ul id="toc-Orientation_vector-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orientation_matrix" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Orientation_matrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.4</span> <span>Orientation matrix</span> </div> </a> <ul id="toc-Orientation_matrix-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orientation_quaternion" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Orientation_quaternion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.5</span> <span>Orientation quaternion</span> </div> </a> <ul id="toc-Orientation_quaternion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Newton's_second_law_in_three_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Newton's_second_law_in_three_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Newton's second law in three dimensions</span> </div> </a> <ul id="toc-Newton's_second_law_in_three_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rigid_system_of_particles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rigid_system_of_particles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Rigid system of particles</span> </div> </a> <ul id="toc-Rigid_system_of_particles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mass_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mass_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Mass properties</span> </div> </a> <ul id="toc-Mass_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Force-torque_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Force-torque_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Force-torque equations</span> </div> </a> <ul id="toc-Force-torque_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotation_in_three_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rotation_in_three_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Rotation in three dimensions</span> </div> </a> <ul id="toc-Rotation_in_three_dimensions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Virtual_work_of_forces_acting_on_a_rigid_body" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Virtual_work_of_forces_acting_on_a_rigid_body"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Virtual work of forces acting on a rigid body</span> </div> </a> <button aria-controls="toc-Virtual_work_of_forces_acting_on_a_rigid_body-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Virtual work of forces acting on a rigid body subsection</span> </button> <ul id="toc-Virtual_work_of_forces_acting_on_a_rigid_body-sublist" class="vector-toc-list"> <li id="toc-Virtual_work" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Virtual_work"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Virtual work</span> </div> </a> <ul id="toc-Virtual_work-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalized_forces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalized_forces"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Generalized forces</span> </div> </a> <ul id="toc-Generalized_forces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-D'Alembert's_form_of_the_principle_of_virtual_work" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#D'Alembert's_form_of_the_principle_of_virtual_work"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>D'Alembert's form of the principle of virtual work</span> </div> </a> <button aria-controls="toc-D'Alembert's_form_of_the_principle_of_virtual_work-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle D'Alembert's form of the principle of virtual work subsection</span> </button> <ul id="toc-D'Alembert's_form_of_the_principle_of_virtual_work-sublist" class="vector-toc-list"> <li id="toc-Static_equilibrium" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Static_equilibrium"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Static equilibrium</span> </div> </a> <ul id="toc-Static_equilibrium-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalized_inertia_forces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalized_inertia_forces"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Generalized inertia forces</span> </div> </a> <ul id="toc-Generalized_inertia_forces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dynamic_equilibrium" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dynamic_equilibrium"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Dynamic equilibrium</span> </div> </a> <ul id="toc-Dynamic_equilibrium-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lagrange's_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lagrange's_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Lagrange's equations</span> </div> </a> <ul id="toc-Lagrange's_equations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Linear_and_angular_momentum" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Linear_and_angular_momentum"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Linear and angular momentum</span> </div> </a> <button aria-controls="toc-Linear_and_angular_momentum-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Linear and angular momentum subsection</span> </button> <ul id="toc-Linear_and_angular_momentum-sublist" class="vector-toc-list"> <li id="toc-System_of_particles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#System_of_particles"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>System of particles</span> </div> </a> <ul id="toc-System_of_particles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rigid_system_of_particles_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rigid_system_of_particles_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Rigid system of particles</span> </div> </a> <ul id="toc-Rigid_system_of_particles_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </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"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </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"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" 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dynamics</span></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. 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class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid" title="Dinàmica del sòlid rígid – Catalan" lang="ca" hreflang="ca" data-title="Dinàmica del sòlid rígid" 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/Starrk%C3%B6rpersimulation" title="Starrkörpersimulation – German" lang="de" hreflang="de" data-title="Starrkörpersimulation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Din%C3%A1mica_de_cuerpos_r%C3%ADgidos" title="Dinámica de cuerpos rígidos – Spanish" lang="es" hreflang="es" data-title="Dinámica de cuerpos rígidos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%DB%8C%D9%86%D8%A7%D9%85%DB%8C%DA%A9_%D8%A7%D8%AC%D8%B3%D8%A7%D9%85_%D8%B5%D9%84%D8%A8" title="دینامیک اجسام صلب – Persian" lang="fa" hreflang="fa" data-title="دینامیک اجسام صلب" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%83%E0%A4%A2%E0%A4%BC_%E0%A4%AA%E0%A4%BF%E0%A4%A3%E0%A5%8D%E0%A4%A1_%E0%A4%97%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80" title="दृढ़ पिण्ड गतिकी – Hindi" lang="hi" hreflang="hi" data-title="दृढ़ पिण्ड गतिकी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Dinamika_benda_tegar" title="Dinamika benda tegar – Indonesian" lang="id" hreflang="id" data-title="Dinamika benda tegar" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ruch_post%C4%99powy_bry%C5%82y_sztywnej" title="Ruch postępowy bryły sztywnej – Polish" lang="pl" hreflang="pl" data-title="Ruch postępowy bryły sztywnej" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Din%C3%A2mica_de_corpo_r%C3%ADgido" title="Dinâmica de corpo rígido – Portuguese" lang="pt" hreflang="pt" data-title="Dinâmica de corpo rígido" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0_%D1%82%D0%B2%D1%91%D1%80%D0%B4%D0%BE%D0%B3%D0%BE_%D1%82%D0%B5%D0%BB%D0%B0" title="Механика твёрдого тела – Russian" lang="ru" hreflang="ru" data-title="Механика твёрдого тела" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kat%C4%B1_cisim_dinami%C4%9Fi" title="Katı cisim dinamiği – Turkish" lang="tr" hreflang="tr" data-title="Katı cisim dinamiği" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a 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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 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.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 the <a href="/wiki/Physics" title="Physics">physical</a> science of <a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">dynamics</a>, <b>rigid-body dynamics</b> studies the movement of <a href="/wiki/Physical_system" title="Physical system">systems</a> of interconnected <a href="/wiki/Physical_object" title="Physical object">bodies</a> under the action of external <a href="/wiki/Force" title="Force">forces</a>. The assumption that the bodies are <i><a href="/wiki/Rigid_body" title="Rigid body">rigid</a></i> (i.e. they do not <a href="/wiki/Deformation_(physics)" title="Deformation (physics)">deform</a> under the action of applied forces) simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of <a href="/wiki/Frame_of_reference" title="Frame of reference">reference frames</a> attached to each body.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> This excludes bodies that display <a href="/wiki/Fluid" title="Fluid">fluid</a>, highly <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a>, and <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">plastic</a> behavior. </p><p>The dynamics of a rigid body system is described by the laws of <a href="/wiki/Kinematics" title="Kinematics">kinematics</a> and by the application of Newton's second law (<a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">kinetics</a>) or their derivative form, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>. The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system, and overall the system itself, as a <a href="/wiki/Time-variant_system" title="Time-variant system">function of time</a>. The formulation and solution of rigid body dynamics is an important tool in the computer simulation of <a href="/wiki/Mechanical_system" class="mw-redirect" title="Mechanical system">mechanical systems</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Planar_rigid_body_dynamics">Planar rigid body dynamics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=1" title="Edit section: Planar rigid body dynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If a system of particles moves parallel to a fixed plane, the system is said to be constrained to planar movement. In this case, Newton's laws (kinetics) for a rigid system of N particles, P<sub><i>i</i></sub>, <i>i</i>=1,...,<i>N</i>, simplify because there is no movement in the <i>k</i> direction. Determine the <a href="/wiki/Resultant_force" title="Resultant force">resultant force</a> and <a href="/wiki/Torque" title="Torque">torque</a> at a reference point <b>R</b>, to obtain <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\sum _{i=1}^{N}m_{i}\mathbf {A} _{i},\quad \mathbf {T} =\sum _{i=1}^{N}(\mathbf {r} _{i}-\mathbf {R} )\times m_{i}\mathbf {A} _{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</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> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</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> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\sum _{i=1}^{N}m_{i}\mathbf {A} _{i},\quad \mathbf {T} =\sum _{i=1}^{N}(\mathbf {r} _{i}-\mathbf {R} )\times m_{i}\mathbf {A} _{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427c0602ef18cd5d68ce4758df14a23993d3410b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.553ex; height:7.343ex;" alt="{\displaystyle \mathbf {F} =\sum _{i=1}^{N}m_{i}\mathbf {A} _{i},\quad \mathbf {T} =\sum _{i=1}^{N}(\mathbf {r} _{i}-\mathbf {R} )\times m_{i}\mathbf {A} _{i},}"></span> </p><p>where <b>r</b><sub>i</sub> denotes the planar trajectory of each particle. </p><p>The <a href="/wiki/Kinematics" title="Kinematics">kinematics</a> of a rigid body yields the formula for the acceleration of the particle P<sub>i</sub> in terms of the position <b>R</b> and acceleration <b>A</b> of the reference particle as well as the angular velocity vector <i><b>ω</b></i> and angular acceleration vector <i><b>α</b></i> of the rigid system of particles as, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} _{i}={\boldsymbol {\alpha }}\times (\mathbf {r} _{i}-\mathbf {R} )+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times (\mathbf {r} _{i}-\mathbf {R} ))+\mathbf {A} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} _{i}={\boldsymbol {\alpha }}\times (\mathbf {r} _{i}-\mathbf {R} )+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times (\mathbf {r} _{i}-\mathbf {R} ))+\mathbf {A} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1e346d5faec580dfe0b2e119af0afe7b58b4dd9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.811ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} _{i}={\boldsymbol {\alpha }}\times (\mathbf {r} _{i}-\mathbf {R} )+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times (\mathbf {r} _{i}-\mathbf {R} ))+\mathbf {A} .}"></span> </p><p>For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along <b>k</b> perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors <b>e</b><sub>i</sub> from the reference point <b>R</b> to a point <b>r</b><sub>i</sub> and the unit vectors <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="{\textstyle \mathbf {t} _{i}=\mathbf {k} \times \mathbf {e} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {t} _{i}=\mathbf {k} \times \mathbf {e} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d018ac0267e984cc8663b6daf5cfdcabb9b136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.214ex; height:2.509ex;" alt="{\textstyle \mathbf {t} _{i}=\mathbf {k} \times \mathbf {e} _{i}}"></span>, so <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} _{i}=\alpha (\Delta r_{i}\mathbf {t} _{i})-\omega ^{2}(\Delta r_{i}\mathbf {e} _{i})+\mathbf {A} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} _{i}=\alpha (\Delta r_{i}\mathbf {t} _{i})-\omega ^{2}(\Delta r_{i}\mathbf {e} _{i})+\mathbf {A} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a139c0293ebb8bd310e9f5ce2c4cc9eb33a1206" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.303ex; height:3.176ex;" alt="{\displaystyle \mathbf {A} _{i}=\alpha (\Delta r_{i}\mathbf {t} _{i})-\omega ^{2}(\Delta r_{i}\mathbf {e} _{i})+\mathbf {A} .}"></span> </p><p>This yields the resultant force on the system as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\alpha \sum _{i=1}^{N}m_{i}\left(\Delta r_{i}\mathbf {t} _{i}\right)-\omega ^{2}\sum _{i=1}^{N}m_{i}\left(\Delta r_{i}\mathbf {e} _{i}\right)+\left(\sum _{i=1}^{N}m_{i}\right)\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>α</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\alpha \sum _{i=1}^{N}m_{i}\left(\Delta r_{i}\mathbf {t} _{i}\right)-\omega ^{2}\sum _{i=1}^{N}m_{i}\left(\Delta r_{i}\mathbf {e} _{i}\right)+\left(\sum _{i=1}^{N}m_{i}\right)\mathbf {A} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1762f3d507e1ce24001e61e673740e51646adce5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:57.53ex; height:7.509ex;" alt="{\displaystyle \mathbf {F} =\alpha \sum _{i=1}^{N}m_{i}\left(\Delta r_{i}\mathbf {t} _{i}\right)-\omega ^{2}\sum _{i=1}^{N}m_{i}\left(\Delta r_{i}\mathbf {e} _{i}\right)+\left(\sum _{i=1}^{N}m_{i}\right)\mathbf {A} ,}"></span> and torque as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {T} ={}&\sum _{i=1}^{N}(m_{i}\Delta r_{i}\mathbf {e} _{i})\times \left(\alpha (\Delta r_{i}\mathbf {t} _{i})-\omega ^{2}(\Delta r_{i}\mathbf {e} _{i})+\mathbf {A} \right)\\{}={}&\left(\sum _{i=1}^{N}m_{i}\Delta r_{i}^{2}\right)\alpha \mathbf {k} +\left(\sum _{i=1}^{N}m_{i}\Delta r_{i}\mathbf {e} _{i}\right)\times \mathbf {A} ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi></mi> <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> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {T} ={}&\sum _{i=1}^{N}(m_{i}\Delta r_{i}\mathbf {e} _{i})\times \left(\alpha (\Delta r_{i}\mathbf {t} _{i})-\omega ^{2}(\Delta r_{i}\mathbf {e} _{i})+\mathbf {A} \right)\\{}={}&\left(\sum _{i=1}^{N}m_{i}\Delta r_{i}^{2}\right)\alpha \mathbf {k} +\left(\sum _{i=1}^{N}m_{i}\Delta r_{i}\mathbf {e} _{i}\right)\times \mathbf {A} ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd9a902e179308b39676fe5218f18847edf18f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:51.619ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {T} ={}&\sum _{i=1}^{N}(m_{i}\Delta r_{i}\mathbf {e} _{i})\times \left(\alpha (\Delta r_{i}\mathbf {t} _{i})-\omega ^{2}(\Delta r_{i}\mathbf {e} _{i})+\mathbf {A} \right)\\{}={}&\left(\sum _{i=1}^{N}m_{i}\Delta r_{i}^{2}\right)\alpha \mathbf {k} +\left(\sum _{i=1}^{N}m_{i}\Delta r_{i}\mathbf {e} _{i}\right)\times \mathbf {A} ,\end{aligned}}}"></span> </p><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="{\textstyle \mathbf {e} _{i}\times \mathbf {e} _{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {e} _{i}\times \mathbf {e} _{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d364c8ca3b6399323c7768edf9864d56ecb18bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.151ex; height:2.509ex;" alt="{\textstyle \mathbf {e} _{i}\times \mathbf {e} _{i}=0}"></span> and <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="{\textstyle \mathbf {e} _{i}\times \mathbf {t} _{i}=\mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {e} _{i}\times \mathbf {t} _{i}=\mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e9fb743937f8a462948e9e63f830f7c3a9eb09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.214ex; height:2.509ex;" alt="{\textstyle \mathbf {e} _{i}\times \mathbf {t} _{i}=\mathbf {k} }"></span> is the unit vector perpendicular to the plane for all of the particles P<sub>i</sub>. </p><p>Use the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> <b>C</b> as the reference point, so these equations for Newton's laws simplify to become <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =M\mathbf {A} ,\quad \mathbf {T} =I_{\textbf {C}}\alpha \mathbf {k} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">C</mtext> </mrow> </mrow> </msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =M\mathbf {A} ,\quad \mathbf {T} =I_{\textbf {C}}\alpha \mathbf {k} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a26121f508b3cdde196a17d60739a7c75579040" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.723ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =M\mathbf {A} ,\quad \mathbf {T} =I_{\textbf {C}}\alpha \mathbf {k} ,}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">M</span> is the total mass and <i>I</i><sub><b>C</b></sub> is the <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> about an axis perpendicular to the movement of the rigid system and through the center of mass. </p> <div class="mw-heading mw-heading2"><h2 id="Rigid_body_in_three_dimensions">Rigid body in three dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=2" title="Edit section: Rigid body in three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Orientation_or_attitude_descriptions">Orientation or attitude descriptions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=3" title="Edit section: Orientation or attitude descriptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Rotation_formalisms_in_three_dimensions" title="Rotation formalisms in three dimensions">Rotation formalisms in three dimensions</a></div> <p>Several methods to describe orientations of a rigid body in three dimensions have been developed. They are summarized in the following sections. </p> <div class="mw-heading mw-heading4"><h4 id="Euler_angles">Euler angles</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=4" title="Edit section: Euler angles"><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/Euler_angles" title="Euler angles">Euler angles</a></div><p> The first attempt to represent an orientation is attributed to <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>. He imagined three reference frames that could rotate one around the other, and realized that by starting with a fixed reference frame and performing three rotations, he could get any other reference frame in the space (using two rotations to fix the vertical axis and another to fix the other two axes). The values of these three rotations are called <a href="/wiki/Euler_angles" title="Euler angles">Euler angles</a>. Commonly, <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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> is used to denote precession, <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 \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></span><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 }"></span> nutation, and <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> intrinsic rotation.</p><ul class="gallery mw-gallery-packed"> <li class="gallerybox" style="width: 219.33333333333px"> <div class="thumb" style="width: 217.33333333333px;"><span typeof="mw:File"><a href="/wiki/File:Euler.png" class="mw-file-description" title="Diagram of the Euler angles"><img alt="Diagram of the Euler angles" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Euler.png/326px-Euler.png" decoding="async" width="218" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Euler.png/489px-Euler.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Euler.png/652px-Euler.png 2x" data-file-width="725" data-file-height="800" /></a></span></div> <div class="gallerytext">Diagram of the Euler angles</div> </li> <li class="gallerybox" style="width: 232px"> <div class="thumb" style="width: 230px;"><span typeof="mw:File"><a href="/wiki/File:Euler_AxisAngle.png" class="mw-file-description" title="Intrinsic rotation of a ball about a fixed axis"><img alt="Intrinsic rotation of a ball about a fixed axis" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Euler_AxisAngle.png/345px-Euler_AxisAngle.png" decoding="async" width="230" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Euler_AxisAngle.png/518px-Euler_AxisAngle.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Euler_AxisAngle.png/690px-Euler_AxisAngle.png 2x" data-file-width="699" data-file-height="729" /></a></span></div> <div class="gallerytext">Intrinsic rotation of a ball about a fixed axis</div> </li> <li class="gallerybox" style="width: 322px"> <div class="thumb" style="width: 320px;"><span typeof="mw:File"><a href="/wiki/File:ToupieEuler.png" class="mw-file-description" title="Motion of a top in the Euler angles"><img alt="Motion of a top in the Euler angles" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/ToupieEuler.png/480px-ToupieEuler.png" decoding="async" width="320" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/ToupieEuler.png/720px-ToupieEuler.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/8e/ToupieEuler.png 2x" data-file-width="800" data-file-height="600" /></a></span></div> <div class="gallerytext">Motion of a top in the Euler angles</div> </li> </ul> <div class="mw-heading mw-heading4"><h4 id="Tait–Bryan_angles"><span id="Tait.E2.80.93Bryan_angles"></span>Tait–Bryan angles</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=5" title="Edit section: Tait–Bryan angles"><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/Euler_angles#Tait–Bryan_angles" title="Euler angles">Euler angles § Tait–Bryan angles</a></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Taitbrianzyx.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Taitbrianzyx.svg/150px-Taitbrianzyx.svg.png" decoding="async" width="150" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Taitbrianzyx.svg/225px-Taitbrianzyx.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Taitbrianzyx.svg/300px-Taitbrianzyx.svg.png 2x" data-file-width="680" data-file-height="665" /></a><figcaption>Tait–Bryan angles, another way to describe orientation</figcaption></figure> <p>These are three angles, also known as yaw, pitch and roll, Navigation angles and Cardan angles. Mathematically they constitute a set of six possibilities inside the twelve possible sets of Euler angles, the ordering being the one best used for describing the orientation of a vehicle such as an airplane. In aerospace engineering they are usually referred to as Euler angles. </p> <div class="mw-heading mw-heading4"><h4 id="Orientation_vector">Orientation vector <span class="anchor" id="Euler_vector"></span><span class="anchor" id="Euler_orientation_vector"></span></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=6" title="Edit section: Orientation vector"><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/Axis-angle_representation" class="mw-redirect" title="Axis-angle representation">Axis-angle representation</a></div> <p>Euler also realized that the composition of two rotations is equivalent to a single rotation about a different fixed axis (<a href="/wiki/Euler%27s_rotation_theorem" title="Euler's rotation theorem">Euler's rotation theorem</a>). Therefore, the composition of the former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed. </p><p>Based on this fact he introduced a vectorial way to describe any rotation, with a vector on the rotation axis and module equal to the value of the angle. Therefore, any orientation can be represented by a rotation vector (also called Euler vector) that leads to it from the reference frame. When used to represent an orientation, the rotation vector is commonly called orientation vector, or attitude vector. </p><p>A similar method, called <a href="/wiki/Axis-angle_representation" class="mw-redirect" title="Axis-angle representation">axis-angle representation</a>, describes a rotation or orientation using a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> aligned with the rotation axis, and a separate value to indicate the angle (see figure). </p> <div class="mw-heading mw-heading4"><h4 id="Orientation_matrix">Orientation matrix</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=7" title="Edit section: Orientation matrix"><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/Rotation_matrix" title="Rotation matrix">Rotation matrix</a></div> <p>With the introduction of matrices the Euler theorems were rewritten. The rotations were described by <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrices</a> referred to as rotation matrices or direction cosine matrices. When used to represent an orientation, a rotation matrix is commonly called orientation matrix, or attitude matrix. </p><p>The above-mentioned Euler vector is the <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvector</a> of a rotation matrix (a rotation matrix has a unique real <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a>). The product of two rotation matrices is the composition of rotations. Therefore, as before, the orientation can be given as the rotation from the initial frame to achieve the frame that we want to describe. </p><p>The <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a> of a non-<a href="/wiki/Symmetry" title="Symmetry">symmetrical</a> object in <i>n</i>-dimensional space is <a href="/wiki/Orthogonal_group" title="Orthogonal group">SO(<i>n</i>)</a> <a href="/wiki/Product_topology" title="Product topology">×</a> <a href="/wiki/Euclidean_space" title="Euclidean space"><b>R</b><sup><i>n</i></sup></a>. Orientation may be visualized by attaching a basis of <a href="/wiki/Tangent_space" title="Tangent space">tangent vectors</a> to an object. The direction in which each vector points determines its orientation. </p> <div class="mw-heading mw-heading4"><h4 id="Orientation_quaternion">Orientation quaternion</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=8" title="Edit section: Orientation quaternion"><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/Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">Quaternions and spatial rotation</a></div> <p>Another way to describe rotations is using <a href="/wiki/Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">rotation quaternions</a>, also called versors. They are equivalent to rotation matrices and rotation vectors. With respect to rotation vectors, they can be more easily converted to and from matrices. When used to represent orientations, rotation quaternions are typically called orientation quaternions or attitude quaternions. </p> <div class="mw-heading mw-heading3"><h3 id="Newton's_second_law_in_three_dimensions"><span id="Newton.27s_second_law_in_three_dimensions"></span>Newton's second law in three dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=9" title="Edit section: Newton's second law in three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To consider rigid body dynamics in three-dimensional space, Newton's second law must be extended to define the relationship between the movement of a rigid body and the system of forces and torques that act on it. </p><p>Newton formulated his second law for a particle as, "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."<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Because Newton generally referred to mass times velocity as the "motion" of a particle, the phrase "change of motion" refers to the mass times acceleration of the particle, and so this law is usually written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =m\mathbf {a} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m\mathbf {a} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93aa54e6c7e8df66d85d06b6eb0b0a2d3ec4ce20" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.768ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =m\mathbf {a} ,}"></span> where <b>F</b> is understood to be the only external force acting on the particle, <i>m</i> is the mass of the particle, and <b>a</b> is its acceleration vector. The extension of Newton's second law to rigid bodies is achieved by considering a rigid system of particles. </p> <div class="mw-heading mw-heading3"><h3 id="Rigid_system_of_particles">Rigid system of particles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=10" title="Edit section: Rigid system of particles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If a system of <i>N</i> particles, P<sub>i</sub>, i=1,...,<i>N</i>, are assembled into a rigid body, then Newton's second law can be applied to each of the particles in the body. If <b>F</b><sub>i</sub> is the external force applied to particle P<sub>i</sub> with mass <i>m</i><sub>i</sub>, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{i}+\sum _{j=1}^{N}\mathbf {F} _{ij}=m_{i}\mathbf {a} _{i},\quad i=1,\ldots ,N,}"> <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>i</mi> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{i}+\sum _{j=1}^{N}\mathbf {F} _{ij}=m_{i}\mathbf {a} _{i},\quad i=1,\ldots ,N,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e0c18eeacb4686829f2754866a81e1c341fede4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.571ex; height:7.676ex;" alt="{\displaystyle \mathbf {F} _{i}+\sum _{j=1}^{N}\mathbf {F} _{ij}=m_{i}\mathbf {a} _{i},\quad i=1,\ldots ,N,}"></span> where <b>F</b><sub>ij</sub> is the internal force of particle P<sub>j</sub> acting on particle P<sub>i</sub> that maintains the constant distance between these particles. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rigid_bodies.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Rigid_bodies.jpg/220px-Rigid_bodies.jpg" decoding="async" width="220" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Rigid_bodies.jpg/330px-Rigid_bodies.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/2/29/Rigid_bodies.jpg 2x" data-file-width="426" data-file-height="492" /></a><figcaption>Human body modelled as a system of rigid bodies of geometrical solids. Representative bones were added for better visualization of the walking person.</figcaption></figure> <p>An important simplification to these force equations is obtained by introducing the <a href="/wiki/Resultant_force" title="Resultant force">resultant force</a> and torque that acts on the rigid system. This resultant force and torque is obtained by choosing one of the particles in the system as a reference point, <b>R</b>, where each of the external forces are applied with the addition of an associated torque. The resultant force <b>F</b> and torque <b>T</b> are given by the formulas, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\sum _{i=1}^{N}\mathbf {F} _{i},\quad \mathbf {T} =\sum _{i=1}^{N}(\mathbf {R} _{i}-\mathbf {R} )\times \mathbf {F} _{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</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> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\sum _{i=1}^{N}\mathbf {F} _{i},\quad \mathbf {T} =\sum _{i=1}^{N}(\mathbf {R} _{i}-\mathbf {R} )\times \mathbf {F} _{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d367b694d2ec9354d0452887a816c9f459d390a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.1ex; height:7.343ex;" alt="{\displaystyle \mathbf {F} =\sum _{i=1}^{N}\mathbf {F} _{i},\quad \mathbf {T} =\sum _{i=1}^{N}(\mathbf {R} _{i}-\mathbf {R} )\times \mathbf {F} _{i},}"></span> where <b>R</b><sub>i</sub> is the vector that defines the position of particle P<sub>i</sub>. </p><p>Newton's second law for a particle combines with these formulas for the resultant force and torque to yield, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\sum _{i=1}^{N}m_{i}\mathbf {a} _{i},\quad \mathbf {T} =\sum _{i=1}^{N}(\mathbf {R} _{i}-\mathbf {R} )\times (m_{i}\mathbf {a} _{i}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</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> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</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> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\sum _{i=1}^{N}m_{i}\mathbf {a} _{i},\quad \mathbf {T} =\sum _{i=1}^{N}(\mathbf {R} _{i}-\mathbf {R} )\times (m_{i}\mathbf {a} _{i}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b02a1883901152253bc9e64a555210337ea1153f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.823ex; height:7.343ex;" alt="{\displaystyle \mathbf {F} =\sum _{i=1}^{N}m_{i}\mathbf {a} _{i},\quad \mathbf {T} =\sum _{i=1}^{N}(\mathbf {R} _{i}-\mathbf {R} )\times (m_{i}\mathbf {a} _{i}),}"></span> where the internal forces <b>F</b><sub><i>ij</i></sub> cancel in pairs. The <a href="/wiki/Kinematics" title="Kinematics">kinematics</a> of a rigid body yields the formula for the acceleration of the particle P<sub>i</sub> in terms of the position <b>R</b> and acceleration <b>a</b> of the reference particle as well as the angular velocity vector ω and angular acceleration vector α of the rigid system of particles as, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{i}=\alpha \times (\mathbf {R} _{i}-\mathbf {R} )+\omega \times (\omega \times (\mathbf {R} _{i}-\mathbf {R} ))+\mathbf {a} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{i}=\alpha \times (\mathbf {R} _{i}-\mathbf {R} )+\omega \times (\omega \times (\mathbf {R} _{i}-\mathbf {R} ))+\mathbf {a} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61448a42bfa530c743d6def306fb46af75e662bd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.446ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} _{i}=\alpha \times (\mathbf {R} _{i}-\mathbf {R} )+\omega \times (\omega \times (\mathbf {R} _{i}-\mathbf {R} ))+\mathbf {a} .}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Mass_properties">Mass properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=11" title="Edit section: Mass properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mass properties of the rigid body are represented by its <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> and <a href="/wiki/Moment_of_inertia" title="Moment of inertia">inertia matrix</a>. Choose the reference point <b>R</b> so that it satisfies the condition <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{N}m_{i}(\mathbf {R} _{i}-\mathbf {R} )=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{N}m_{i}(\mathbf {R} _{i}-\mathbf {R} )=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eeebb0a4e1d8d7d7b02c8266f80f479bdadfd11" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.946ex; height:7.343ex;" alt="{\displaystyle \sum _{i=1}^{N}m_{i}(\mathbf {R} _{i}-\mathbf {R} )=0,}"></span> </p><p>then it is known as the center of mass of the system. </p><p>The inertia matrix [I<sub>R</sub>] of the system relative to the reference point <b>R</b> is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [I_{R}]=\sum _{i=1}^{N}m_{i}\left(\mathbf {I} \left(\mathbf {S} _{i}^{\textsf {T}}\mathbf {S} _{i}\right)-\mathbf {S} _{i}\mathbf {S} _{i}^{\textsf {T}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [I_{R}]=\sum _{i=1}^{N}m_{i}\left(\mathbf {I} \left(\mathbf {S} _{i}^{\textsf {T}}\mathbf {S} _{i}\right)-\mathbf {S} _{i}\mathbf {S} _{i}^{\textsf {T}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d212d70cb031aa272e6fef3690439553a1a07450" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.643ex; height:7.343ex;" alt="{\displaystyle [I_{R}]=\sum _{i=1}^{N}m_{i}\left(\mathbf {I} \left(\mathbf {S} _{i}^{\textsf {T}}\mathbf {S} _{i}\right)-\mathbf {S} _{i}\mathbf {S} _{i}^{\textsf {T}}\right),}"></span> </p><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 {S} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926b0315a51772356bb6ec4c269f87e390989944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle \mathbf {S} _{i}}"></span> is the column vector <span class="texhtml"><b>R</b><sub>i</sub> − <b>R</b></span>; <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 {S} _{i}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} _{i}^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d53f2a737342097c01569b5ecba028b99e77ed0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.837ex; height:3.176ex;" alt="{\displaystyle \mathbf {S} _{i}^{\textsf {T}}}"></span> is its transpose, and <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 {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {I} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a458c8aeb096ce732abf346ae8edf3e4f53a126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }"></span> is the 3 by 3 identity matrix. </p><p><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 {S} _{i}^{\textsf {T}}\mathbf {S} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} _{i}^{\textsf {T}}\mathbf {S} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d85c6340f01b6b5318489627ed47ee272e258763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.122ex; height:3.176ex;" alt="{\displaystyle \mathbf {S} _{i}^{\textsf {T}}\mathbf {S} _{i}}"></span> is the scalar product of <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 {S} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926b0315a51772356bb6ec4c269f87e390989944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle \mathbf {S} _{i}}"></span> with itself, while <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 {S} _{i}\mathbf {S} _{i}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} _{i}\mathbf {S} _{i}^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ff75eee88e2532d11c2441a769bf6962c37fd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.122ex; height:3.176ex;" alt="{\displaystyle \mathbf {S} _{i}\mathbf {S} _{i}^{\textsf {T}}}"></span> is the tensor product of <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 {S} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926b0315a51772356bb6ec4c269f87e390989944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle \mathbf {S} _{i}}"></span> with itself. </p> <div class="mw-heading mw-heading3"><h3 id="Force-torque_equations">Force-torque equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=12" title="Edit section: Force-torque equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using the center of mass and inertia matrix, the force and torque equations for a single rigid body take the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =m\mathbf {a} ,\quad \mathbf {T} =[I_{R}]\alpha +\omega \times [I_{R}]\omega ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mi>α</mi> <mo>+</mo> <mi>ω</mi> <mo>×</mo> <mo stretchy="false">[</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mi>ω</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m\mathbf {a} ,\quad \mathbf {T} =[I_{R}]\alpha +\omega \times [I_{R}]\omega ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04cb3ac3348dc67b9f1d15aa50079857a33a8dcf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.735ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} =m\mathbf {a} ,\quad \mathbf {T} =[I_{R}]\alpha +\omega \times [I_{R}]\omega ,}"></span> and are known as Newton's second law of motion for a rigid body. </p><p>The dynamics of an interconnected system of rigid bodies, <span class="texhtml"><i>B</i><sub><i>i</i></sub></span>, <span class="texhtml"><i>j</i> = 1, ..., <i>M</i></span>, is formulated by isolating each rigid body and introducing the interaction forces. The resultant of the external and interaction forces on each body, yields the force-torque equations <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{j}=m_{j}\mathbf {a} _{j},\quad \mathbf {T} _{j}=[I_{R}]_{j}\alpha _{j}+\omega _{j}\times [I_{R}]_{j}\omega _{j},\quad j=1,\ldots ,M.}"> <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>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>×</mo> <mo stretchy="false">[</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{j}=m_{j}\mathbf {a} _{j},\quad \mathbf {T} _{j}=[I_{R}]_{j}\alpha _{j}+\omega _{j}\times [I_{R}]_{j}\omega _{j},\quad j=1,\ldots ,M.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb22fbde3580ac19702abca9caee8df4f187f576" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.118ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} _{j}=m_{j}\mathbf {a} _{j},\quad \mathbf {T} _{j}=[I_{R}]_{j}\alpha _{j}+\omega _{j}\times [I_{R}]_{j}\omega _{j},\quad j=1,\ldots ,M.}"></span> </p><p>Newton's formulation yields 6<i>M</i> equations that define the dynamics of a system of <i>M</i> rigid bodies.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Rotation_in_three_dimensions">Rotation in three dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=13" title="Edit section: Rotation in three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A rotating object, whether under the influence of torques or not, may exhibit the behaviours of <a href="/wiki/Precession" title="Precession">precession</a> and <a href="/wiki/Nutation" title="Nutation">nutation</a>. The fundamental equation describing the behavior of a rotating solid body is <a href="/wiki/Euler%27s_equation_of_motion" class="mw-redirect" title="Euler's equation of motion">Euler's equation of motion</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}={\frac {D\mathbf {L} }{Dt}}={\frac {d\mathbf {L} }{dt}}+{\boldsymbol {\omega }}\times \mathbf {L} ={\frac {d(I{\boldsymbol {\omega }})}{dt}}+{\boldsymbol {\omega }}\times {I{\boldsymbol {\omega }}}=I{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times {I{\boldsymbol {\omega }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <mo>=</mo> <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 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 class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mo>=</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}={\frac {D\mathbf {L} }{Dt}}={\frac {d\mathbf {L} }{dt}}+{\boldsymbol {\omega }}\times \mathbf {L} ={\frac {d(I{\boldsymbol {\omega }})}{dt}}+{\boldsymbol {\omega }}\times {I{\boldsymbol {\omega }}}=I{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times {I{\boldsymbol {\omega }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5a8badfbc980772529b28cfe3ea6ef1ee430df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:60.821ex; height:5.843ex;" alt="{\displaystyle {\boldsymbol {\tau }}={\frac {D\mathbf {L} }{Dt}}={\frac {d\mathbf {L} }{dt}}+{\boldsymbol {\omega }}\times \mathbf {L} ={\frac {d(I{\boldsymbol {\omega }})}{dt}}+{\boldsymbol {\omega }}\times {I{\boldsymbol {\omega }}}=I{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times {I{\boldsymbol {\omega }}}}"></span> where the <a href="/wiki/Pseudovector" title="Pseudovector">pseudovectors</a> <b>τ</b> and <b>L</b> are, respectively, the <a href="/wiki/Torque" title="Torque">torques</a> on the body and its <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>, the scalar <i>I</i> is its <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>, the vector <b>ω</b> is its angular velocity, the vector <b>α</b> is its angular acceleration, D is the differential in an inertial reference frame and d is the differential in a relative reference frame fixed with the body. </p><p>The solution to this equation when there is no applied torque is discussed in the articles <a href="/wiki/Euler%27s_equation_of_motion" class="mw-redirect" title="Euler's equation of motion">Euler's equation of motion</a> and <a href="/wiki/Poinsot%27s_ellipsoid" title="Poinsot's ellipsoid">Poinsot's ellipsoid</a>. </p><p>It follows from Euler's equation that a torque <b>τ</b> applied perpendicular to the axis of rotation, and therefore perpendicular to <b>L</b>, results in a rotation about an axis perpendicular to both <b>τ</b> and <b>L</b>. This motion is called <i>precession</i>. The angular velocity of precession <b>Ω</b><sub>P</sub> is given by the <a href="/wiki/Cross_product" title="Cross product">cross product</a>:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2015)">citation needed</span></a></i>]</sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}={\boldsymbol {\Omega }}_{\mathrm {P} }\times \mathbf {L} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}={\boldsymbol {\Omega }}_{\mathrm {P} }\times \mathbf {L} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6887b5e83576087b8f141908c3a027eb962dc600" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.895ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\tau }}={\boldsymbol {\Omega }}_{\mathrm {P} }\times \mathbf {L} .}"></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gyroscope_precession.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Gyroscope_precession.gif/220px-Gyroscope_precession.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/82/Gyroscope_precession.gif 1.5x" data-file-width="300" data-file-height="300" /></a><figcaption>Precession of a <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a></figcaption></figure> <p>Precession can be demonstrated by placing a spinning top with its axis horizontal and supported loosely (frictionless toward precession) at one end. Instead of falling, as might be expected, the top appears to defy gravity by remaining with its axis horizontal, when the other end of the axis is left unsupported and the free end of the axis slowly describes a circle in a horizontal plane, the resulting precession turning. This effect is explained by the above equations. The torque on the top is supplied by a couple of forces: gravity acting downward on the device's centre of mass, and an equal force acting upward to support one end of the device. The rotation resulting from this torque is not downward, as might be intuitively expected, causing the device to fall, but perpendicular to both the gravitational torque (horizontal and perpendicular to the axis of rotation) and the axis of rotation (horizontal and outwards from the point of support), i.e., about a vertical axis, causing the device to rotate slowly about the supporting point. </p><p>Under a constant torque of magnitude <i>τ</i>, the speed of precession <i>Ω</i><sub>P</sub> is inversely proportional to <i>L</i>, the magnitude of its angular momentum: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ={\mathit {\Omega }}_{\mathrm {P} }L\sin \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">Ω</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mi>L</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ={\mathit {\Omega }}_{\mathrm {P} }L\sin \theta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83feb43e05c26be25916b5cc256a22ffeeb0eac2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.266ex; height:2.509ex;" alt="{\displaystyle \tau ={\mathit {\Omega }}_{\mathrm {P} }L\sin \theta ,}"></span> where <i>θ</i> is the angle between the vectors <b>Ω</b><sub>P</sub> and <b>L</b>. Thus, if the top's spin slows down (for example, due to friction), its angular momentum decreases and so the rate of precession increases. This continues until the device is unable to rotate fast enough to support its own weight, when it stops precessing and falls off its support, mostly because friction against precession cause another precession that goes to cause the fall. </p><p>By convention, these three vectors - torque, spin, and precession - are all oriented with respect to each other according to the <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand rule</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Virtual_work_of_forces_acting_on_a_rigid_body">Virtual work of forces acting on a rigid body</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=14" title="Edit section: Virtual work of forces acting on a rigid body"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An alternate formulation of rigid body dynamics that has a number of convenient features is obtained by considering the <a href="/wiki/Virtual_work" title="Virtual work">virtual work</a> of forces acting on a rigid body. </p><p>The virtual work of forces acting at various points on a single rigid body can be calculated using the velocities of their point of application and the <a href="/wiki/Resultant_force" title="Resultant force">resultant force and torque</a>. To see this, let the forces <b>F</b><sub>1</sub>, <b>F</b><sub>2</sub> ... <b>F</b><sub><i>n</i></sub> act on the points <b>R</b><sub>1</sub>, <b>R</b><sub>2</sub> ... <b>R</b><sub><i>n</i></sub> in a rigid body. </p><p>The trajectories of <b>R</b><sub><i>i</i></sub>, <span class="texhtml"><i>i</i> = 1, ..., <i>n</i></span> are defined by the movement of the rigid body. The velocity of the points <b>R</b><sub><i>i</i></sub> along their trajectories are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V} _{i}={\boldsymbol {\omega }}\times (\mathbf {R} _{i}-\mathbf {R} )+\mathbf {V} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} _{i}={\boldsymbol {\omega }}\times (\mathbf {R} _{i}-\mathbf {R} )+\mathbf {V} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666f1a6403fe2bcfc884d6c19acb47d90d65fc45" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.389ex; height:2.843ex;" alt="{\displaystyle \mathbf {V} _{i}={\boldsymbol {\omega }}\times (\mathbf {R} _{i}-\mathbf {R} )+\mathbf {V} ,}"></span> where <b>ω</b> is the angular velocity vector of the body. </p> <div class="mw-heading mw-heading3"><h3 id="Virtual_work">Virtual work</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=15" title="Edit section: Virtual work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Work is computed from the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of each force with the displacement of its point of contact <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3f8b0f46e6e55d58d67c73c0fb3105f05414ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.083ex; height:6.843ex;" alt="{\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}.}"></span> If the trajectory of a rigid body is defined by a set of <a href="/wiki/Generalized_coordinates" title="Generalized coordinates">generalized coordinates</a> <span class="texhtml"><i>q</i><sub><i>j</i></sub></span>, <span class="texhtml"><i>j</i> = 1, ..., <i>m</i></span>, then the virtual displacements <span class="texhtml"><i>δ</i><b>r</b><sub>i</sub></span> are given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}}\delta q_{j}=\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}\delta q_{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}}\delta q_{j}=\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}\delta q_{j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd39eb7eefb474eaf220ee0bc4a702d53c5945e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.343ex; height:7.176ex;" alt="{\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}}\delta q_{j}=\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}\delta q_{j}.}"></span> The virtual work of this system of forces acting on the body in terms of the generalized coordinates becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\mathbf {F} _{1}\cdot \left(\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{1}}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)+\dots +\mathbf {F} _{n}\cdot \left(\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{n}}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\mathbf {F} _{1}\cdot \left(\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{1}}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)+\dots +\mathbf {F} _{n}\cdot \left(\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{n}}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664f819be796b2574eaff040224e20bfb624a1b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:55.44ex; height:7.676ex;" alt="{\displaystyle \delta W=\mathbf {F} _{1}\cdot \left(\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{1}}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)+\dots +\mathbf {F} _{n}\cdot \left(\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{n}}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)}"></span> </p><p>or collecting the coefficients of <span class="texhtml"><i>δq<sub>j</sub></i></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{1}}}\right)\delta q_{1}+\dots +\left(\sum _{1=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{m}}}\right)\delta q_{m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{1}}}\right)\delta q_{1}+\dots +\left(\sum _{1=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{m}}}\right)\delta q_{m}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214a6323e22af25a49da16609b24620245ff5c34" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.658ex; height:7.509ex;" alt="{\displaystyle \delta W=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{1}}}\right)\delta q_{1}+\dots +\left(\sum _{1=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{m}}}\right)\delta q_{m}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Generalized_forces">Generalized forces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=16" title="Edit section: Generalized forces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For simplicity consider a trajectory of a rigid body that is specified by a single generalized coordinate q, such as a rotation angle, then the formula becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}}}\right)\delta q=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial ({\boldsymbol {\omega }}\times (\mathbf {R} _{i}-\mathbf {R} )+\mathbf {V} )}{\partial {\dot {q}}}}\right)\delta q.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </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> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <mi>q</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> </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> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <mi>q</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}}}\right)\delta q=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial ({\boldsymbol {\omega }}\times (\mathbf {R} _{i}-\mathbf {R} )+\mathbf {V} )}{\partial {\dot {q}}}}\right)\delta q.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700375171a7e560adfc388d5a6e9df99a0a19489" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:66.27ex; height:7.509ex;" alt="{\displaystyle \delta W=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}}}\right)\delta q=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial ({\boldsymbol {\omega }}\times (\mathbf {R} _{i}-\mathbf {R} )+\mathbf {V} )}{\partial {\dot {q}}}}\right)\delta q.}"></span> </p><p>Introduce the resultant force <b>F</b> and torque <b>T</b> so this equation takes the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\left(\mathbf {F} \cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}}}+\mathbf {T} \cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}}}\right)\delta q.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </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 class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </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> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <mi>q</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\left(\mathbf {F} \cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}}}+\mathbf {T} \cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}}}\right)\delta q.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d6c88843bf69c44ce323bb17d0f1892c1b686f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.893ex; height:6.176ex;" alt="{\displaystyle \delta W=\left(\mathbf {F} \cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}}}+\mathbf {T} \cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}}}\right)\delta q.}"></span> </p><p>The quantity <i>Q</i> defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\mathbf {F} \cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}}}+\mathbf {T} \cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </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 class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\mathbf {F} \cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}}}+\mathbf {T} \cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/298137edfc43383451cb7d53a41f6c669df1229c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.321ex; height:5.843ex;" alt="{\displaystyle Q=\mathbf {F} \cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}}}+\mathbf {T} \cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}}},}"></span> </p><p>is known as the <a href="/wiki/Generalized_forces" title="Generalized forces">generalized force</a> associated with the virtual displacement δq. This formula generalizes to the movement of a rigid body defined by more than one generalized coordinate, that is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\sum _{j=1}^{m}Q_{j}\delta q_{j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\sum _{j=1}^{m}Q_{j}\delta q_{j},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a679d873e64385d0b6211b80a7ce3c53aeba369" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.715ex; height:7.176ex;" alt="{\displaystyle \delta W=\sum _{j=1}^{m}Q_{j}\delta q_{j},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{j}=\mathbf {F} \cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}_{j}}}+\mathbf {T} \cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{j}=\mathbf {F} \cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}_{j}}}+\mathbf {T} \cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4b730e85542e125203bb0f39381881812ccff2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.91ex; height:6.343ex;" alt="{\displaystyle Q_{j}=\mathbf {F} \cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}_{j}}}+\mathbf {T} \cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.}"></span> </p><p>It is useful to note that conservative forces such as gravity and spring forces are derivable from a potential function <span class="texhtml"><i>V</i>(<i>q</i><sub>1</sub>, ..., <i>q</i><sub><i>n</i></sub>)</span>, known as a <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>. In this case the generalized forces are given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{j}=-{\frac {\partial V}{\partial q_{j}}},\quad j=1,\ldots ,m.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{j}=-{\frac {\partial V}{\partial q_{j}}},\quad j=1,\ldots ,m.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4308ff5cfb56cc37b94786a305f4ee3023d3ce89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.197ex; height:6.176ex;" alt="{\displaystyle Q_{j}=-{\frac {\partial V}{\partial q_{j}}},\quad j=1,\ldots ,m.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="D'Alembert's_form_of_the_principle_of_virtual_work"><span id="D.27Alembert.27s_form_of_the_principle_of_virtual_work"></span>D'Alembert's form of the principle of virtual work</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=17" title="Edit section: D'Alembert's form of the principle of virtual work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The equations of motion for a mechanical system of rigid bodies can be determined using D'Alembert's form of the principle of virtual work. The principle of virtual work is used to study the static equilibrium of a system of rigid bodies, however by introducing acceleration terms in Newton's laws this approach is generalized to define dynamic equilibrium. </p> <div class="mw-heading mw-heading3"><h3 id="Static_equilibrium">Static equilibrium</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=18" title="Edit section: Static equilibrium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The static equilibrium of a mechanical system rigid bodies is defined by the condition that the virtual work of the applied forces is zero for any virtual displacement of the system. This is known as the <i>principle of virtual work.</i><sup id="cite_ref-Torby1984_5-0" class="reference"><a href="#cite_note-Torby1984-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is <i>Q</i><sub><i>i</i></sub>=0. </p><p>Let a mechanical system be constructed from <span class="texhtml mvar" style="font-style:italic;">n</span> rigid bodies, B<sub><i>i</i></sub>, <span class="texhtml"><i>i</i> = 1, ..., <i>n</i></span>, and let the resultant of the applied forces on each body be the force-torque pairs, <span class="texhtml"><b>F</b><sub><i>i</i></sub></span> and <span class="texhtml"><b>T</b><sub><i>i</i></sub></span>, <span class="texhtml"><i>i</i> = 1, ..., <i>n</i></span>. Notice that these applied forces do not include the reaction forces where the bodies are connected. Finally, assume that the velocity <span class="texhtml"><b>V</b><sub><i>i</i></sub></span> and angular velocities <span class="texhtml"><i><b>ω</b></i><sub><i>i</i></sub></span>, <span class="texhtml"><i>i</i> = 1, ..., <i>n</i></span>, for each rigid body, are defined by a single generalized coordinate q. Such a system of rigid bodies is said to have one <a href="/wiki/Degree_of_freedom_(mechanics)" class="mw-redirect" title="Degree of freedom (mechanics)">degree of freedom</a>. </p><p>The virtual work of the forces and torques, <span class="texhtml"><b>F</b><sub><i>i</i></sub></span> and <span class="texhtml"><b>T</b><sub><i>i</i></sub></span>, applied to this one degree of freedom system is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\boldsymbol {\omega }}_{i}}{\partial {\dot {q}}}}\right)\delta q=Q\delta q,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </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> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <mi>q</mi> <mo>=</mo> <mi>Q</mi> <mi>δ</mi> <mi>q</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\boldsymbol {\omega }}_{i}}{\partial {\dot {q}}}}\right)\delta q=Q\delta q,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e44163221edb0e5c11f6026697e5501bbb04bd1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.889ex; height:6.843ex;" alt="{\displaystyle \delta W=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\boldsymbol {\omega }}_{i}}{\partial {\dot {q}}}}\right)\delta q=Q\delta q,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\boldsymbol {\omega }}_{i}}{\partial {\dot {q}}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </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> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\boldsymbol {\omega }}_{i}}{\partial {\dot {q}}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a765c7283c8ed365074f043606b1d0102ad01538" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.07ex; height:6.843ex;" alt="{\displaystyle Q=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\boldsymbol {\omega }}_{i}}{\partial {\dot {q}}}}\right),}"></span> is the generalized force acting on this one degree of freedom system. </p><p>If the mechanical system is defined by m generalized coordinates, <span class="texhtml"><i>q</i><sub><i>j</i></sub></span>, <span class="texhtml"><i>j</i> = 1, ..., <i>m</i></span>, then the system has m degrees of freedom and the virtual work is given by, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\sum _{j=1}^{m}Q_{j}\delta q_{j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\sum _{j=1}^{m}Q_{j}\delta q_{j},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a679d873e64385d0b6211b80a7ce3c53aeba369" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.715ex; height:7.176ex;" alt="{\displaystyle \delta W=\sum _{j=1}^{m}Q_{j}\delta q_{j},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{j}=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\boldsymbol {\omega }}_{i}}{\partial {\dot {q}}_{j}}}\right),\quad j=1,\ldots ,m.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{j}=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\boldsymbol {\omega }}_{i}}{\partial {\dot {q}}_{j}}}\right),\quad j=1,\ldots ,m.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c868f553e7c0828fae1ef13c45003042f6839dc2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.034ex; height:7.509ex;" alt="{\displaystyle Q_{j}=\sum _{i=1}^{n}\left(\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}+\mathbf {T} _{i}\cdot {\frac {\partial {\boldsymbol {\omega }}_{i}}{\partial {\dot {q}}_{j}}}\right),\quad j=1,\ldots ,m.}"></span> is the generalized force associated with the generalized coordinate <span class="texhtml"><i>q</i><sub><i>j</i></sub></span>. The principle of virtual work states that static equilibrium occurs when these generalized forces acting on the system are zero, that is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{j}=0,\quad j=1,\ldots ,m.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{j}=0,\quad j=1,\ldots ,m.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7858fae475a047df5a6ff2f68ef2d0d3749f5db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.45ex; height:2.843ex;" alt="{\displaystyle Q_{j}=0,\quad j=1,\ldots ,m.}"></span> </p><p>These <span class="texhtml mvar" style="font-style:italic;">m</span> equations define the static equilibrium of the system of rigid bodies. </p> <div class="mw-heading mw-heading3"><h3 id="Generalized_inertia_forces">Generalized inertia forces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=19" title="Edit section: Generalized inertia forces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a single rigid body which moves under the action of a resultant force <b>F</b> and torque <b>T</b>, with one degree of freedom defined by the generalized coordinate <i>q</i>. Assume the reference point for the resultant force and torque is the center of mass of the body, then the generalized inertia force <span class="texhtml"><i>Q*</i></span> associated with the generalized coordinate <span class="texhtml mvar" style="font-style:italic;">q</span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{*}=-(M\mathbf {A} )\cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}}}-\left([I_{R}]{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times [I_{R}]{\boldsymbol {\omega }}\right)\cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </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> <mo stretchy="false">[</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">[</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{*}=-(M\mathbf {A} )\cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}}}-\left([I_{R}]{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times [I_{R}]{\boldsymbol {\omega }}\right)\cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df57abd9197f239cfcfa19770f34cfeeb699b1e3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.101ex; height:5.843ex;" alt="{\displaystyle Q^{*}=-(M\mathbf {A} )\cdot {\frac {\partial \mathbf {V} }{\partial {\dot {q}}}}-\left([I_{R}]{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times [I_{R}]{\boldsymbol {\omega }}\right)\cdot {\frac {\partial {\boldsymbol {\omega }}}{\partial {\dot {q}}}}.}"></span> </p><p>This inertia force can be computed from the kinetic energy of the rigid body, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\tfrac {1}{2}}M\mathbf {V} \cdot \mathbf {V} +{\tfrac {1}{2}}{\boldsymbol {\omega }}\cdot [I_{R}]{\boldsymbol {\omega }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">[</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\tfrac {1}{2}}M\mathbf {V} \cdot \mathbf {V} +{\tfrac {1}{2}}{\boldsymbol {\omega }}\cdot [I_{R}]{\boldsymbol {\omega }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9746f56c6d59a6412cccc00201d6b3846f7af724" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.512ex; height:3.509ex;" alt="{\displaystyle T={\tfrac {1}{2}}M\mathbf {V} \cdot \mathbf {V} +{\tfrac {1}{2}}{\boldsymbol {\omega }}\cdot [I_{R}]{\boldsymbol {\omega }},}"></span> by using the formula <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{*}=-\left({\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}}}-{\frac {\partial T}{\partial q}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</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 class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>q</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{*}=-\left({\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}}}-{\frac {\partial T}{\partial q}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5011d97e0cbee68fa732ce8991a61549118888e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.954ex; height:6.176ex;" alt="{\displaystyle Q^{*}=-\left({\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}}}-{\frac {\partial T}{\partial q}}\right).}"></span> </p><p>A system of <span class="texhtml mvar" style="font-style:italic;">n</span> rigid bodies with m generalized coordinates has the kinetic energy <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=\sum _{i=1}^{n}\left({\tfrac {1}{2}}M\mathbf {V} _{i}\cdot \mathbf {V} _{i}+{\tfrac {1}{2}}{\boldsymbol {\omega }}_{i}\cdot [I_{R}]{\boldsymbol {\omega }}_{i}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>M</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">[</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">]</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=\sum _{i=1}^{n}\left({\tfrac {1}{2}}M\mathbf {V} _{i}\cdot \mathbf {V} _{i}+{\tfrac {1}{2}}{\boldsymbol {\omega }}_{i}\cdot [I_{R}]{\boldsymbol {\omega }}_{i}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cab5ab8718b6844af72b9fa8b6ba4324a1152d60" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.969ex; height:6.843ex;" alt="{\displaystyle T=\sum _{i=1}^{n}\left({\tfrac {1}{2}}M\mathbf {V} _{i}\cdot \mathbf {V} _{i}+{\tfrac {1}{2}}{\boldsymbol {\omega }}_{i}\cdot [I_{R}]{\boldsymbol {\omega }}_{i}\right),}"></span> which can be used to calculate the m generalized inertia forces<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> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{j}^{*}=-\left({\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}\right),\quad j=1,\ldots ,m.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{j}^{*}=-\left({\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}\right),\quad j=1,\ldots ,m.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f16308afd5d1ea4567b0e47ebb0ee3f37b1a1a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.97ex; height:7.509ex;" alt="{\displaystyle Q_{j}^{*}=-\left({\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}\right),\quad j=1,\ldots ,m.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Dynamic_equilibrium">Dynamic equilibrium</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=20" title="Edit section: Dynamic equilibrium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system. Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\left(Q_{1}+Q_{1}^{*}\right)\delta q_{1}+\dots +\left(Q_{m}+Q_{m}^{*}\right)\delta q_{m}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\left(Q_{1}+Q_{1}^{*}\right)\delta q_{1}+\dots +\left(Q_{m}+Q_{m}^{*}\right)\delta q_{m}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd5c1b4fe5c2de2649857b9721d370261a23b38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.68ex; height:3.009ex;" alt="{\displaystyle \delta W=\left(Q_{1}+Q_{1}^{*}\right)\delta q_{1}+\dots +\left(Q_{m}+Q_{m}^{*}\right)\delta q_{m}=0,}"></span> for any set of virtual displacements <span class="texhtml"><i>δq</i><sub><i>j</i></sub></span>. This condition yields <span class="texhtml mvar" style="font-style:italic;">m</span> equations, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{j}+Q_{j}^{*}=0,\quad j=1,\ldots ,m,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{j}+Q_{j}^{*}=0,\quad j=1,\ldots ,m,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abdc1026adbba50c42e5d87d85441336a6f2d9f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.183ex; height:3.176ex;" alt="{\displaystyle Q_{j}+Q_{j}^{*}=0,\quad j=1,\ldots ,m,}"></span> which can also be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}=Q_{j},\quad j=1,\ldots ,m.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}=Q_{j},\quad j=1,\ldots ,m.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588bfb13eedb4ccb044296df1df54a9a409cfb53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:36.561ex; height:6.343ex;" alt="{\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}=Q_{j},\quad j=1,\ldots ,m.}"></span> The result is a set of m equations of motion that define the dynamics of the rigid body system. </p> <div class="mw-heading mw-heading3"><h3 id="Lagrange's_equations"><span id="Lagrange.27s_equations"></span>Lagrange's equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=21" title="Edit section: Lagrange's equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the generalized forces Q<sub><i>j</i></sub> are derivable from a potential energy <span class="texhtml"><i>V</i>(<i>q</i><sub>1</sub>, ..., <i>q</i><sub><i>m</i></sub>)</span>, then these equations of motion take the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}=-{\frac {\partial V}{\partial q_{j}}},\quad j=1,\ldots ,m.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}=-{\frac {\partial V}{\partial q_{j}}},\quad j=1,\ldots ,m.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd686304a9c80e9af12e8f88db50df653ab7906" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:39.723ex; height:6.343ex;" alt="{\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}=-{\frac {\partial V}{\partial q_{j}}},\quad j=1,\ldots ,m.}"></span> </p><p>In this case, introduce the <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a>, <span class="texhtml"><i>L</i> = <i>T</i> − <i>V</i></span>, so these equations of motion become <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}-{\frac {\partial L}{\partial q_{j}}}=0\quad j=1,\ldots ,m.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}-{\frac {\partial L}{\partial q_{j}}}=0\quad j=1,\ldots ,m.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21b186aa6c6042e89c4f950413bac44a96000b1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.942ex; height:6.343ex;" alt="{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}-{\frac {\partial L}{\partial q_{j}}}=0\quad j=1,\ldots ,m.}"></span> These are known as <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrange's equations of motion</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Linear_and_angular_momentum">Linear and angular momentum</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=22" title="Edit section: Linear and angular momentum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="System_of_particles">System of particles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=23" title="Edit section: System of particles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The linear and angular momentum of a rigid system of particles is formulated by measuring the position and velocity of the particles relative to the center of mass. Let the system of particles P<sub>i</sub>, <span class="texhtml"><i>i</i> = 1, ..., <i>n</i></span> be located at the coordinates <b>r</b><sub><i>i</i></sub> and velocities <b>v</b><sub><i>i</i></sub>. Select a reference point <b>R</b> and compute the relative position and velocity vectors, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}=\left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {R} ,\quad \mathbf {v} _{i}={\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {V} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}=\left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {R} ,\quad \mathbf {v} _{i}={\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {V} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/596822f0cf74caddd7b22d744d530faf7d2be438" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.017ex; height:5.509ex;" alt="{\displaystyle \mathbf {r} _{i}=\left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {R} ,\quad \mathbf {v} _{i}={\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {V} .}"></span> </p><p>The total linear and angular momentum vectors relative to the reference point <b>R</b> are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} ={\frac {d}{dt}}\left(\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\right)+\left(\sum _{i=1}^{n}m_{i}\right)\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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} ={\frac {d}{dt}}\left(\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\right)+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edeb445aaf99af1181fcab5266f04d6f8b815bf7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.225ex; height:7.509ex;" alt="{\displaystyle \mathbf {p} ={\frac {d}{dt}}\left(\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\right)+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} ,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\times {\frac {d}{dt}}\left(\mathbf {r} _{i}-\mathbf {R} \right)+\left(\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\right)\times \mathbf {V} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\times {\frac {d}{dt}}\left(\mathbf {r} _{i}-\mathbf {R} \right)+\left(\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\right)\times \mathbf {V} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0eddfc8baf6959f02c443fed85202311a0f8a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.457ex; height:7.509ex;" alt="{\displaystyle \mathbf {L} =\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\times {\frac {d}{dt}}\left(\mathbf {r} _{i}-\mathbf {R} \right)+\left(\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\right)\times \mathbf {V} .}"></span> </p><p>If <b>R</b> is chosen as the center of mass these equations simplify to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =M\mathbf {V} ,\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\times {\frac {d}{dt}}\left(\mathbf {r} _{i}-\mathbf {R} \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</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> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =M\mathbf {V} ,\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\times {\frac {d}{dt}}\left(\mathbf {r} _{i}-\mathbf {R} \right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98504f3d24d209f8a11597a8b7d5acee03f15ad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.34ex; height:6.843ex;" alt="{\displaystyle \mathbf {p} =M\mathbf {V} ,\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}\left(\mathbf {r} _{i}-\mathbf {R} \right)\times {\frac {d}{dt}}\left(\mathbf {r} _{i}-\mathbf {R} \right).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Rigid_system_of_particles_2">Rigid system of particles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=24" title="Edit section: Rigid system of particles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To specialize these formulas to a rigid body, assume the particles are rigidly connected to each other so P<sub>i</sub>, i=1,...,n are located by the coordinates <b>r</b><sub>i</sub> and velocities <b>v</b><sub>i</sub>. Select a reference point <b>R</b> and compute the relative position and velocity vectors, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}=(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {R} ,\quad \mathbf {v} _{i}=\omega \times (\mathbf {r} _{i}-\mathbf {R} )+\mathbf {V} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}=(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {R} ,\quad \mathbf {v} _{i}=\omega \times (\mathbf {r} _{i}-\mathbf {R} )+\mathbf {V} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a90412b356e44c087004a14a7aa1c364ac00f50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.412ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} _{i}=(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {R} ,\quad \mathbf {v} _{i}=\omega \times (\mathbf {r} _{i}-\mathbf {R} )+\mathbf {V} ,}"></span> where ω is the angular velocity of the system.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The <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> of this rigid system measured relative to the center of mass <b>R</b> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} ,\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times \mathbf {v} _{i}=\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times (\omega \times (\mathbf {r} _{i}-\mathbf {R} )).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</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> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} ,\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times \mathbf {v} _{i}=\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times (\omega \times (\mathbf {r} _{i}-\mathbf {R} )).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1231ff3858788338ec01fc7879a7fe8a6b41dad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:81.878ex; height:7.509ex;" alt="{\displaystyle \mathbf {p} =\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} ,\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times \mathbf {v} _{i}=\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times (\omega \times (\mathbf {r} _{i}-\mathbf {R} )).}"></span> </p><p>These equations simplify to become, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =M\mathbf {V} ,\quad \mathbf {L} =[I_{R}]\omega ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mi>ω</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =M\mathbf {V} ,\quad \mathbf {L} =[I_{R}]\omega ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ef7db20a056f0716e9e45d93ee4c733871f5fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.998ex; height:2.843ex;" alt="{\displaystyle \mathbf {p} =M\mathbf {V} ,\quad \mathbf {L} =[I_{R}]\omega ,}"></span> where M is the total mass of the system and [I<sub>R</sub>] is the <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> matrix defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [I_{R}]=-\sum _{i=1}^{n}m_{i}[r_{i}-R][r_{i}-R],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">[</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>R</mi> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>R</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [I_{R}]=-\sum _{i=1}^{n}m_{i}[r_{i}-R][r_{i}-R],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe90de29df41040a0385f1f1f79d2f99af97e03" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.812ex; height:6.843ex;" alt="{\displaystyle [I_{R}]=-\sum _{i=1}^{n}m_{i}[r_{i}-R][r_{i}-R],}"></span> where [r<sub><i>i</i></sub> − R] is the skew-symmetric matrix constructed from the vector <b>r</b><sub><i>i</i></sub> − <b>R</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=25" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>For the analysis of robotic systems</li> <li>For the biomechanical analysis of animals, humans or humanoid systems</li> <li>For the analysis of space objects</li> <li>For the understanding of strange motions of rigid bodies.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li>For the design and development of dynamics-based sensors, such as gyroscopic sensors.</li> <li>For the design and development of various stability enhancement applications in automobiles.</li> <li>For improving the graphics of video games which involves rigid bodies</li></ul> <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=Rigid_body_dynamics&action=edit&section=26" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></li> <li><a href="/wiki/Analytical_dynamics" class="mw-redirect" title="Analytical dynamics">Analytical dynamics</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">Dynamics (mechanics)</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/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li> <li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a></li> <li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li> <li><a href="/wiki/Rigid_body" title="Rigid body">Rigid body</a></li> <li><a href="/wiki/Rigid_transformation" title="Rigid transformation">Rigid transformation</a></li> <li><a href="/wiki/Rigid_rotor" title="Rigid rotor">Rigid rotor</a></li> <li><a href="/wiki/Soft-body_dynamics" title="Soft-body dynamics">Soft-body dynamics</a></li> <li><a href="/wiki/Multibody_system" title="Multibody system">Multibody system</a></li> <li><a href="/wiki/Polhode" title="Polhode">Polhode</a></li> <li><a href="/wiki/Herpolhode" title="Herpolhode">Herpolhode</a></li> <li><a href="/wiki/Precession" title="Precession">Precession</a></li> <li><a href="/wiki/Poinsot%27s_ellipsoid" title="Poinsot's ellipsoid">Poinsot's ellipsoid</a></li> <li><a href="/wiki/Gyroscope" title="Gyroscope">Gyroscope</a></li> <li><a href="/wiki/Physics_engine" title="Physics engine">Physics engine</a></li> <li><a href="/wiki/Physics_processing_unit" title="Physics processing unit">Physics processing unit</a></li> <li><a href="/wiki/Physics_Abstraction_Layer" title="Physics Abstraction Layer">Physics Abstraction Layer</a> – Unified multibody simulator</li> <li><a href="/wiki/RigidChips" title="RigidChips">RigidChips</a> – Japanese rigid-body simulator</li> <li><a href="/wiki/Euler%27s_Equation" class="mw-redirect" title="Euler's Equation">Euler's Equation</a></li></ul> </div> <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=Rigid_body_dynamics&action=edit&section=27" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">B. Paul, Kinematics and Dynamics of Planar Machinery, Prentice-Hall, NJ, 1979</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">L. W. Tsai, Robot Analysis: The mechanics of serial and parallel manipulators, John-Wiley, NY, 1999.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Encyclopædia Britannica, <a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/371907/mechanics/77534/Newtons-laws-of-motion-and-equilibrium">Newtons laws of motion</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">K. J. Waldron and G. L. Kinzel, <a rel="nofollow" class="external text" href="https://www.amazon.com/Kinematics-Dynamics-Design-Machinery-Waldron/dp/0471583995">Kinematics and Dynamics, and Design of Machinery</a>, 2nd Ed., John Wiley and Sons, 2004.</span> </li> <li id="cite_note-Torby1984-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Torby1984_5-0">^</a></b></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="CITEREFTorby1984" class="citation book cs1">Torby, Bruce (1984). "Energy Methods". <i>Advanced Dynamics for Engineers</i>. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-063366-4" title="Special:BookSources/0-03-063366-4"><bdi>0-03-063366-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Energy+Methods&rft.btitle=Advanced+Dynamics+for+Engineers&rft.place=United+States+of+America&rft.series=HRW+Series+in+Mechanical+Engineering&rft.pub=CBS+College+Publishing&rft.date=1984&rft.isbn=0-03-063366-4&rft.aulast=Torby&rft.aufirst=Bruce&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARigid+body+dynamics" 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">T. R. Kane and D. A. Levinson, <a rel="nofollow" class="external text" href="https://www.amazon.com/Dynamics-Theory-Applications-Mechanical-Engineering/dp/0070378460">Dynamics, Theory and Applications</a>, McGraw-Hill, NY, 2005.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarionThornton1995" class="citation book cs1">Marion, JB; Thornton, ST (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/classicaldynamic00mari_0"><i>Classical Dynamics of Systems and Particles</i></a></span> (4th ed.). Thomson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-097302-3" title="Special:BookSources/0-03-097302-3"><bdi>0-03-097302-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Dynamics+of+Systems+and+Particles&rft.edition=4th&rft.pub=Thomson&rft.date=1995&rft.isbn=0-03-097302-3&rft.aulast=Marion&rft.aufirst=JB&rft.au=Thornton%2C+ST&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fclassicaldynamic00mari_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARigid+body+dynamics" class="Z3988"></span>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSymon1971" class="citation book cs1">Symon, KR (1971). <i>Mechanics</i> (3rd ed.). Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-07392-7" title="Special:BookSources/0-201-07392-7"><bdi>0-201-07392-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=1971&rft.isbn=0-201-07392-7&rft.aulast=Symon&rft.aufirst=KR&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARigid+body+dynamics" class="Z3988"></span>.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTenenbaum2004" class="citation book cs1">Tenenbaum, RA (2004). <a rel="nofollow" class="external text" href="https://archive.org/details/springer_10.1007-b97616"><i>Fundamentals of Applied Dynamics</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-00887-X" title="Special:BookSources/0-387-00887-X"><bdi>0-387-00887-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=Fundamentals+of+Applied+Dynamics&rft.pub=Springer&rft.date=2004&rft.isbn=0-387-00887-X&rft.aulast=Tenenbaum&rft.aufirst=RA&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fspringer_10.1007-b97616&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARigid+body+dynamics" class="Z3988"></span>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGomezHernandez-GomezMarquina2012" class="citation journal cs1">Gomez, R W; Hernandez-Gomez, J J; Marquina, V (25 July 2012). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/236030807">"A jumping cylinder on an inclined plane"</a>. <i>Eur. J. Phys</i>. <b>33</b> (5). IOP: 1359–1365. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1204.0600">1204.0600</a></span>. <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/2012EJPh...33.1359G">2012EJPh...33.1359G</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%2F0143-0807%2F33%2F5%2F1359">10.1088/0143-0807/33/5/1359</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:55442794">55442794</a><span class="reference-accessdate">. Retrieved <span class="nowrap">25 April</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Eur.+J.+Phys.&rft.atitle=A+jumping+cylinder+on+an+inclined+plane&rft.volume=33&rft.issue=5&rft.pages=1359-1365&rft.date=2012-07-25&rft_id=info%3Aarxiv%2F1204.0600&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A55442794%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0143-0807%2F33%2F5%2F1359&rft_id=info%3Abibcode%2F2012EJPh...33.1359G&rft.aulast=Gomez&rft.aufirst=R+W&rft.au=Hernandez-Gomez%2C+J+J&rft.au=Marquina%2C+V&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F236030807&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARigid+body+dynamics" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=28" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>E. Leimanis (1965). <i>The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point.</i> (<a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer</a>, New York).</li> <li>W. B. Heard (2006). <i>Rigid Body Mechanics: Mathematics, Physics and Applications.</i> (<a href="/wiki/Wiley-VCH" title="Wiley-VCH">Wiley-VCH</a>).</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rigid_body_dynamics&action=edit&section=29" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.d6.com/users/checker/dynamics.htm">Chris Hecker's Rigid Body Dynamics Information</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070312162343/http://www.d6.com/users/checker/dynamics.htm">Archived</a> 12 March 2007 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://www.cs.cmu.edu/~baraff/sigcourse/index.html">Physically Based Modeling: Principles and Practice</a></li> <li><a rel="nofollow" class="external text" href="http://www.digitalrune.com/KnowledgeBase/Overview/tabid/471/Default.aspx">DigitalRune Knowledge Base</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081120191316/http://www.digitalrune.com/KnowledgeBase/Overview/tabid/471/Default.aspx">Archived</a> 20 November 2008 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> contains a master thesis and a collection of resources about rigid body dynamics.</li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/klein_-_line_geom_rigid_bodies.pdf">F. Klein, "Note on the connection between line geometry and the mechanics of rigid bodies"</a> (English translation)</li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/klein_-_screws.pdf">F. Klein, "On Sir Robert Ball's theory of screws"</a> (English translation)</li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/cotton_-_cayley_geometry_and_rigid_motions.pdf">E. Cotton, "Application of Cayley geometry to the geometric study of the displacement of a solid around a fixed point"</a> (English translation)</li></ul> <p class="mw-empty-elt"> </p> <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:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q2037529#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85040318">United States</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Dynamique des corps rigides"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11982035n">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Dynamique des corps rigides"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11982035n">BnF data</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007567969305171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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