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class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Description_of_objects_and_their_motion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Description_of_objects_and_their_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Description of objects and their motion</span> </div> </a> <button aria-controls="toc-Description_of_objects_and_their_motion-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 Description of objects and their motion subsection</span> </button> <ul id="toc-Description_of_objects_and_their_motion-sublist" class="vector-toc-list"> <li id="toc-Kinematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kinematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Kinematics</span> </div> </a> <ul id="toc-Kinematics-sublist" class="vector-toc-list"> <li id="toc-Velocity_and_speed" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Velocity_and_speed"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Velocity and speed</span> </div> </a> <ul id="toc-Velocity_and_speed-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Acceleration" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Acceleration"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Acceleration</span> </div> </a> <ul id="toc-Acceleration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frames_of_reference" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Frames_of_reference"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Frames of reference</span> </div> </a> <ul id="toc-Frames_of_reference-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Newtonian_mechanics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Newtonian_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Newtonian mechanics</span> </div> </a> <button aria-controls="toc-Newtonian_mechanics-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 Newtonian mechanics subsection</span> </button> <ul id="toc-Newtonian_mechanics-sublist" class="vector-toc-list"> <li id="toc-Work_and_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Work_and_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Work and energy</span> </div> </a> <ul id="toc-Work_and_energy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lagrangian_mechanics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lagrangian_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Lagrangian mechanics</span> </div> </a> <ul id="toc-Lagrangian_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hamiltonian_mechanics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hamiltonian_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Hamiltonian mechanics</span> </div> </a> <ul id="toc-Hamiltonian_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limits_of_validity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limits_of_validity"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Limits of validity</span> </div> </a> <button aria-controls="toc-Limits_of_validity-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 Limits of validity subsection</span> </button> <ul id="toc-Limits_of_validity-sublist" class="vector-toc-list"> <li id="toc-Newtonian_approximation_to_special_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Newtonian_approximation_to_special_relativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Newtonian approximation to special relativity</span> </div> </a> <ul id="toc-Newtonian_approximation_to_special_relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classical_approximation_to_quantum_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_approximation_to_quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Classical approximation to quantum mechanics</span> </div> </a> <ul id="toc-Classical_approximation_to_quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</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" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Classical mechanics</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. Available in 112 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-112" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">112 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Klassieke_meganika" title="Klassieke meganika – Afrikaans" lang="af" hreflang="af" data-title="Klassieke meganika" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Klassische_Mechanik" title="Klassische Mechanik – Alemannic" lang="gsw" hreflang="gsw" data-title="Klassische Mechanik" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%AA%D9%82%D9%84%D9%8A%D8%AF%D9%8A%D8%A9" title="ميكانيكا تقليدية – Arabic" lang="ar" hreflang="ar" data-title="ميكانيكا تقليدية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Mecanica_clasica" title="Mecanica clasica – Aragonese" lang="an" hreflang="an" data-title="Mecanica clasica" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%A7%E0%A7%8D%E0%A7%B0%E0%A7%81%E0%A6%AA%E0%A6%A6%E0%A7%80_%E0%A6%AC%E0%A6%B2%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8" title="ধ্ৰুপদী বলবিজ্ঞান – Assamese" lang="as" hreflang="as" data-title="ধ্ৰুপদী বলবিজ্ঞান" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Mec%C3%A1nica_cl%C3%A1sica" title="Mecánica clásica – Asturian" lang="ast" hreflang="ast" data-title="Mecánica clásica" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Klassik_mexanika" title="Klassik mexanika – Azerbaijani" lang="az" hreflang="az" data-title="Klassik mexanika" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%A9%D9%84%D8%A7%D8%B3%DB%8C%DA%A9_%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9" title="کلاسیک مکانیک – South Azerbaijani" lang="azb" hreflang="azb" data-title="کلاسیک مکانیک" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9A%E0%A6%BF%E0%A6%B0%E0%A6%BE%E0%A6%AF%E0%A6%BC%E0%A6%A4_%E0%A6%AC%E0%A6%B2%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8" title="চিরায়ত বলবিজ্ঞান – Bangla" lang="bn" hreflang="bn" data-title="চিরায়ত বলবিজ্ঞান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/K%C3%B3%CD%98-ti%C3%A1n_le%CC%8Dk-ha%CC%8Dk" title="Kó͘-tián le̍k-ha̍k – Minnan" lang="nan" hreflang="nan" data-title="Kó͘-tián le̍k-ha̍k" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-map-bms mw-list-item"><a href="https://map-bms.wikipedia.org/wiki/Mekanika_klasik" title="Mekanika klasik – Banyumasan" lang="jv-x-bms" hreflang="jv-x-bms" data-title="Mekanika klasik" data-language-autonym="Basa Banyumasan" data-language-local-name="Banyumasan" class="interlanguage-link-target"><span>Basa Banyumasan</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%81%D0%B8%D0%BA_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Классик механика – Bashkir" lang="ba" hreflang="ba" data-title="Классик механика" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Класічная механіка – Belarusian" lang="be" hreflang="be" data-title="Класічная механіка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%BB%D1%8F%D1%81%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%BC%D1%8D%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Клясычная мэханіка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Клясычная мэханіка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B2%E0%A4%BE%E0%A4%B8%E0%A4%BF%E0%A4%95%E0%A4%B2_%E0%A4%AE%E0%A5%88%E0%A4%95%E0%A5%87%E0%A4%A8%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%B8" title="क्लासिकल मैकेनिक्स – Bhojpuri" lang="bh" hreflang="bh" data-title="क्लासिकल मैकेनिक्स" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Класическа механика – Bulgarian" lang="bg" hreflang="bg" data-title="Класическа механика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Klasi%C4%8Dna_mehanika" title="Klasična mehanika – Bosnian" lang="bs" hreflang="bs" data-title="Klasična mehanika" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D2%BA%D1%83%D0%BD%D0%B3%D0%B0%D0%B4%D0%B0%D0%B3_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Һунгадаг механика – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Һунгадаг механика" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mec%C3%A0nica_cl%C3%A0ssica" title="Mecànica clàssica – Catalan" lang="ca" hreflang="ca" data-title="Mecànica clàssica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%81%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Классикăлла механика – Chuvash" lang="cv" hreflang="cv" data-title="Классикăлла механика" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Klasick%C3%A1_mechanika" title="Klasická mechanika – Czech" lang="cs" hreflang="cs" data-title="Klasická mechanika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Mecaneg_glasurol" title="Mecaneg glasurol – Welsh" lang="cy" hreflang="cy" data-title="Mecaneg glasurol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Klassisk_mekanik" title="Klassisk mekanik – Danish" lang="da" hreflang="da" data-title="Klassisk mekanik" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Klassische_Mechanik" title="Klassische Mechanik – German" lang="de" hreflang="de" data-title="Klassische Mechanik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Klassikaline_mehaanika" title="Klassikaline mehaanika – Estonian" lang="et" hreflang="et" data-title="Klassikaline mehaanika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%BB%CE%B1%CF%83%CE%B9%CE%BA%CE%AE_%CE%BC%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE" title="Κλασική μηχανική – Greek" lang="el" hreflang="el" data-title="Κλασική μηχανική" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Mec%C3%A1nica_cl%C3%A1sica" title="Mecánica clásica – Spanish" lang="es" hreflang="es" data-title="Mecánica clásica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Klasika_mekaniko" title="Klasika mekaniko – Esperanto" lang="eo" hreflang="eo" data-title="Klasika mekaniko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Mekanika_klasiko" title="Mekanika klasiko – Basque" lang="eu" hreflang="eu" data-title="Mekanika klasiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%DA%A9%D9%84%D8%A7%D8%B3%DB%8C%DA%A9" title="مکانیک کلاسیک – Persian" lang="fa" hreflang="fa" data-title="مکانیک کلاسیک" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Classical_mechanics" title="Classical mechanics – Fiji Hindi" lang="hif" hreflang="hif" data-title="Classical mechanics" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9canique_newtonienne" title="Mécanique newtonienne – French" lang="fr" hreflang="fr" data-title="Mécanique newtonienne" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Meicnic_Newton" title="Meicnic Newton – Irish" lang="ga" hreflang="ga" data-title="Meicnic Newton" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Mec%C3%A1nica_cl%C3%A1sica" title="Mecánica clásica – Galician" lang="gl" hreflang="gl" data-title="Mecánica clásica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%A0%EC%A0%84%EC%97%AD%ED%95%99" title="고전역학 – Korean" lang="ko" hreflang="ko" data-title="고전역학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%A1%D5%BD%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1" title="Դասական մեխանիկա – Armenian" lang="hy" hreflang="hy" data-title="Դասական մեխանիկա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9A%E0%A4%BF%E0%A4%B0%E0%A4%B8%E0%A4%AE%E0%A5%8D%E0%A4%AE%E0%A4%A4_%E0%A4%AF%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80" title="चिरसम्मत यांत्रिकी – Hindi" lang="hi" hreflang="hi" data-title="चिरसम्मत यांत्रिकी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Klasi%C4%8Dna_mehanika" title="Klasična mehanika – Croatian" lang="hr" hreflang="hr" data-title="Klasična mehanika" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Klasika_mekaniko" title="Klasika mekaniko – Ido" lang="io" hreflang="io" data-title="Klasika mekaniko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Mekanika_klasik" title="Mekanika klasik – Indonesian" lang="id" hreflang="id" data-title="Mekanika klasik" 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-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Mechanica_classic" title="Mechanica classic – Interlingua" lang="ia" hreflang="ia" data-title="Mechanica classic" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-ie mw-list-item"><a href="https://ie.wikipedia.org/wiki/Mecanica_classic" title="Mecanica classic – Interlingue" lang="ie" hreflang="ie" data-title="Mecanica classic" data-language-autonym="Interlingue" data-language-local-name="Interlingue" class="interlanguage-link-target"><span>Interlingue</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/S%C3%ADgild_aflfr%C3%A6%C3%B0i" title="Sígild aflfræði – Icelandic" lang="is" hreflang="is" data-title="Sígild aflfræði" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Meccanica_classica" title="Meccanica classica – Italian" lang="it" hreflang="it" data-title="Meccanica classica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%9B%D7%A0%D7%99%D7%A7%D7%94_%D7%A7%D7%9C%D7%90%D7%A1%D7%99%D7%AA" title="מכניקה קלאסית – Hebrew" lang="he" hreflang="he" data-title="מכניקה קלאסית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%A4%E0%B2%BF%E0%B2%B5%E0%B2%BF%E0%B2%9C%E0%B3%8D%E0%B2%9E%E0%B2%BE%E0%B2%A8" title="ಗತಿವಿಜ್ಞಾನ – Kannada" lang="kn" hreflang="kn" data-title="ಗತಿವಿಜ್ಞಾನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%9A%E1%83%90%E1%83%A1%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%9B%E1%83%94%E1%83%A5%E1%83%90%E1%83%9C%E1%83%98%E1%83%99%E1%83%90" title="კლასიკური მექანიკა – Georgian" lang="ka" hreflang="ka" data-title="კლასიკური მექანიკა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%81%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Классикалық механика – Kazakh" lang="kk" hreflang="kk" data-title="Классикалық механика" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Umakanika_kawaida" title="Umakanika kawaida – Swahili" lang="sw" hreflang="sw" data-title="Umakanika kawaida" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/M%C3%A9kanik_newtonyenn" title="Mékanik newtonyenn – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Mékanik newtonyenn" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Mechanica_Newtoniana" title="Mechanica Newtoniana – Latin" lang="la" hreflang="la" data-title="Mechanica Newtoniana" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Klasisk%C4%81_meh%C4%81nika" title="Klasiskā mehānika – Latvian" lang="lv" hreflang="lv" data-title="Klasiskā mehānika" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Klasikin%C4%97_mechanika" title="Klasikinė mechanika – Lithuanian" lang="lt" hreflang="lt" data-title="Klasikinė mechanika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Mecanica_clasica" title="Mecanica clasica – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Mecanica clasica" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Mecanega_classega" title="Mecanega classega – Lombard" lang="lmo" hreflang="lmo" data-title="Mecanega classega" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Klasszikus_mechanika" title="Klasszikus mechanika – Hungarian" lang="hu" hreflang="hu" data-title="Klasszikus mechanika" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D0%B8%D1%87%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Класична механика – Macedonian" lang="mk" hreflang="mk" data-title="Класична механика" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%89%E0%B4%A6%E0%B4%BE%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%AC%E0%B4%B2%E0%B4%A4%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82" title="ഉദാത്തബലതന്ത്രം – Malayalam" lang="ml" hreflang="ml" data-title="ഉദാത്തബലതന്ത്രം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Mekkanika_klassika" title="Mekkanika klassika – Maltese" lang="mt" hreflang="mt" data-title="Mekkanika klassika" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%AD%E0%A4%BF%E0%A4%9C%E0%A4%BE%E0%A4%A4_%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%95%E0%A5%80" title="अभिजात यामिकी – Marathi" lang="mr" hreflang="mr" data-title="अभिजात यामिकी" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%99%E1%83%9A%E1%83%90%E1%83%A1%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%9B%E1%83%94%E1%83%A5%E1%83%90%E1%83%9C%E1%83%98%E1%83%99%E1%83%90" title="კლასიკური მექანიკა – Mingrelian" lang="xmf" hreflang="xmf" data-title="კლასიკური მექანიკა" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%A7%D9%84%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D9%83%D9%84%D8%A7%D8%B3%D9%8A%D9%83%D9%8A%D9%87" title="الميكانيكا الكلاسيكيه – Egyptian Arabic" lang="arz" hreflang="arz" data-title="الميكانيكا الكلاسيكيه" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Mekanik_klasik" title="Mekanik klasik – Malay" lang="ms" hreflang="ms" data-title="Mekanik klasik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A1%D0%BE%D0%BD%D0%B3%D0%BE%D0%B4%D0%BE%D0%B3_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA" title="Сонгодог механик – Mongolian" lang="mn" hreflang="mn" data-title="Сонгодог механик" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%82%E1%80%94%E1%80%B9%E1%80%91%E1%80%9D%E1%80%84%E1%80%BA_%E1%80%99%E1%80%80%E1%80%B9%E1%80%80%E1%80%84%E1%80%BA%E1%80%B8%E1%80%94%E1%80%85%E1%80%BA" title="ဂန္ထဝင် မက္ကင်းနစ် – Burmese" lang="my" hreflang="my" data-title="ဂန္ထဝင် မက္ကင်းနစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Klassieke_mechanica" title="Klassieke mechanica – Dutch" lang="nl" hreflang="nl" data-title="Klassieke mechanica" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%A4%E5%85%B8%E5%8A%9B%E5%AD%A6" title="古典力学 – Japanese" lang="ja" hreflang="ja" data-title="古典力学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ce mw-list-item"><a href="https://ce.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%81%D0%B8%D0%BA%D0%B0%D0%BD_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Классикан механика – Chechen" lang="ce" hreflang="ce" data-title="Классикан механика" data-language-autonym="Нохчийн" data-language-local-name="Chechen" class="interlanguage-link-target"><span>Нохчийн</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Klassisk_mekanikk" title="Klassisk mekanikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Klassisk mekanikk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Klassisk_mekanikk" title="Klassisk mekanikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Klassisk mekanikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Mecanica_classica" title="Mecanica classica – Occitan" lang="oc" hreflang="oc" data-title="Mecanica classica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Klassik_mexanika" title="Klassik mexanika – Uzbek" lang="uz" hreflang="uz" data-title="Klassik mexanika" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9F%E0%A8%95%E0%A8%B8%E0%A8%BE%E0%A8%B2%E0%A9%80_%E0%A8%AE%E0%A8%95%E0%A9%88%E0%A8%A8%E0%A8%95%E0%A9%80" title="ਟਕਸਾਲੀ ਮਕੈਨਕੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਟਕਸਾਲੀ ਮਕੈਨਕੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%A9%D9%84%D8%A7%D8%B3%DB%8C%DA%A9%D9%84_%D9%85%DA%A9%DB%8C%D9%86%DA%A9%D8%B3" title="کلاسیکل مکینکس – Western Punjabi" lang="pnb" hreflang="pnb" data-title="کلاسیکل مکینکس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Klasikal_mikianix" title="Klasikal mikianix – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Klasikal mikianix" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Klassische_Mechanik" title="Klassische Mechanik – Low German" lang="nds" hreflang="nds" data-title="Klassische Mechanik" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Mechanika_klasyczna" title="Mechanika klasyczna – Polish" lang="pl" hreflang="pl" data-title="Mechanika klasyczna" 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/Mec%C3%A2nica_cl%C3%A1ssica" title="Mecânica clássica – Portuguese" lang="pt" hreflang="pt" data-title="Mecânica clássica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Klassik_mexanika" title="Klassik mexanika – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Klassik mexanika" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Mecanic%C4%83_clasic%C4%83" title="Mecanică clasică – Romanian" lang="ro" hreflang="ro" data-title="Mecanică clasică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%96%D1%87%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Класічна механіка – Rusyn" lang="rue" hreflang="rue" data-title="Класічна механіка" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%81%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Классическая механика – Russian" lang="ru" hreflang="ru" data-title="Классическая механика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9E%D0%BB%D0%BE%D2%95%D1%83%D1%80%D0%B1%D1%83%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Олоҕурбут механика – Yakut" lang="sah" hreflang="sah" data-title="Олоҕурбут механика" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Mekanika_klasike" title="Mekanika klasike – Albanian" lang="sq" hreflang="sq" data-title="Mekanika klasike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%B8%E0%B7%8A%E0%B6%B7%E0%B7%8F%E0%B7%80%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B6%BA_%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%8A%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%80" title="සම්භාව්‍යය යාන්ත්‍ර විද්‍යාව – Sinhala" lang="si" hreflang="si" data-title="සම්භාව්‍යය යාන්ත්‍ර විද්‍යාව" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Classical_mechanics" title="Classical mechanics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Classical mechanics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%DA%AA%D9%84%D8%A7%D8%B3%D9%8A%DA%AA%D9%84_%D9%85%D9%8A%DA%AA%D8%A7%D9%86%D9%8A%D8%A7%D8%AA" title="ڪلاسيڪل ميڪانيات – Sindhi" lang="sd" hreflang="sd" data-title="ڪلاسيڪل ميڪانيات" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Klasick%C3%A1_mechanika" title="Klasická mechanika – Slovak" lang="sk" hreflang="sk" data-title="Klasická mechanika" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Klasi%C4%8Dna_mehanika" title="Klasična mehanika – Slovenian" lang="sl" hreflang="sl" data-title="Klasična mehanika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%DB%8C%DA%A9%D8%A7%D9%86%DB%8C%DA%A9%DB%8C_%DA%A9%D9%84%D8%A7%D8%B3%DB%8C%DA%A9" title="میکانیکی کلاسیک – Central Kurdish" lang="ckb" hreflang="ckb" data-title="میکانیکی کلاسیک" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D0%B8%D1%87%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Класична механика – Serbian" lang="sr" hreflang="sr" data-title="Класична механика" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Klasi%C4%8Dna_mehanika" title="Klasična mehanika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Klasična mehanika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Klassinen_mekaniikka" title="Klassinen mekaniikka – Finnish" lang="fi" hreflang="fi" data-title="Klassinen mekaniikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Klassisk_mekanik" title="Klassisk mekanik – Swedish" lang="sv" hreflang="sv" data-title="Klassisk mekanik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Klasikong_mekanika" title="Klasikong mekanika – Tagalog" lang="tl" hreflang="tl" data-title="Klasikong mekanika" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AE%B0%E0%AE%AA%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%A8%E0%AF%8D%E0%AE%A4_%E0%AE%B5%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%AF%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="மரபார்ந்த விசையியல் – Tamil" lang="ta" hreflang="ta" data-title="மரபார்ந்த விசையியல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%81%D0%B8%D0%BA_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Классик механика – Tatar" lang="tt" hreflang="tt" data-title="Классик механика" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B0%BE%E0%B0%82%E0%B0%AA%E0%B1%8D%E0%B0%B0%E0%B0%A6%E0%B0%BE%E0%B0%AF_%E0%B0%AF%E0%B0%BE%E0%B0%82%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%BF%E0%B0%95%E0%B0%B6%E0%B0%BE%E0%B0%B8%E0%B1%8D%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%82" title="సాంప్రదాయ యాంత్రికశాస్త్రం – Telugu" lang="te" hreflang="te" data-title="సాంప్రదాయ యాంత్రికశాస్త్రం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B8%94%E0%B8%B1%E0%B9%89%E0%B8%87%E0%B9%80%E0%B8%94%E0%B8%B4%E0%B8%A1" title="กลศาสตร์ดั้งเดิม – Thai" lang="th" hreflang="th" data-title="กลศาสตร์ดั้งเดิม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D0%B8_%D0%BA%D0%BB%D0%B0%D1%81%D1%81%D0%B8%D0%BA%D3%A3" title="Механикаи классикӣ – Tajik" lang="tg" hreflang="tg" data-title="Механикаи классикӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Klasik_mekanik" title="Klasik mekanik – Turkish" lang="tr" hreflang="tr" data-title="Klasik mekanik" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tyv mw-list-item"><a href="https://tyv.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%81%D0%B8%D0%BA%D1%82%D0%B8%D0%B3_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Классиктиг механика – Tuvinian" lang="tyv" hreflang="tyv" data-title="Классиктиг механика" data-language-autonym="Тыва дыл" data-language-local-name="Tuvinian" class="interlanguage-link-target"><span>Тыва дыл</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D0%B8%D1%87%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Класична механіка – Ukrainian" lang="uk" hreflang="uk" data-title="Класична механіка" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%A9%D9%84%D8%A7%D8%B3%DB%8C%DA%A9%DB%8C_%D9%85%DB%8C%DA%A9%D8%A7%D9%86%DB%8C%D8%A7%D8%AA" title="کلاسیکی میکانیات – Urdu" lang="ur" hreflang="ur" data-title="کلاسیکی میکانیات" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Klassine_mehanik" title="Klassine mehanik – Veps" lang="vep" hreflang="vep" data-title="Klassine mehanik" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_c%E1%BB%95_%C4%91i%E1%BB%83n" title="Cơ học cổ điển – Vietnamese" lang="vi" hreflang="vi" data-title="Cơ học cổ điển" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%B6%93%E5%85%B8%E5%8A%9B%E5%AD%B8" title="經典力學 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="經典力學" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Mekanika_klasika" title="Mekanika klasika – Waray" lang="war" hreflang="war" data-title="Mekanika klasika" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Classical+mechanics%22">"Classical mechanics"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Classical+mechanics%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Classical+mechanics%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Classical+mechanics%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Classical+mechanics%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Classical+mechanics%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">July 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Orbital_motion.gif" class="mw-file-description"><img alt="animation of orbital velocity and centripetal acceleration" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Orbital_motion.gif/261px-Orbital_motion.gif" decoding="async" width="261" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/4e/Orbital_motion.gif 1.5x" data-file-width="300" data-file-height="300" /></a><figcaption>Diagram of orbital motion of a satellite around the Earth, showing perpendicular velocity and acceleration (force) vectors, represented through a classical 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-left:0.9em;padding-right:0.9em;"><a class="mw-selflink selflink">Classical mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">F</mtext> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ad0a6d6780c3abc5247abd82bd8a2249d56ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.318ex; height:5.509ex;" alt="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"></span><div class="sidebar-caption" style="font-size:90%;padding:0.6em 0;font-style:italic;"><a href="/wiki/Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</a></div></td></tr><tr><th class="sidebar-heading" style="font-weight: bold; display:block;margin-bottom:1.0em;"> <div class="hlist"> <ul><li><a href="/wiki/History_of_classical_mechanics" title="History of classical mechanics">History</a></li> <li><a href="/wiki/Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline</a></li> <li><a href="/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">Textbooks</a></li></ul> </div></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Applied_mechanics" title="Applied mechanics">Applied</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li> <li><a href="/wiki/Analytical_dynamics" class="mw-redirect" title="Analytical dynamics">Dynamics</a></li> <li><a href="/wiki/Classical_field_theory" title="Classical field theory">Field theory</a></li> <li><a href="/wiki/Kinematics" title="Kinematics">Kinematics</a></li> <li><a href="/wiki/Kinetics_(physics)" title="Kinetics (physics)">Kinetics</a></li> <li><a href="/wiki/Statics" title="Statics">Statics</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Acceleration" title="Acceleration">Acceleration</a></li> <li><a href="/wiki/Angular_momentum" title="Angular momentum">Angular momentum</a></li> <li><a href="/wiki/Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li> <li><a href="/wiki/D%27Alembert%27s_principle" title="D'Alembert's principle">D'Alembert's principle</a></li> <li><a href="/wiki/Energy" title="Energy">Energy</a> <ul><li><a href="/wiki/Kinetic_energy#Newtonian_kinetic_energy" title="Kinetic energy">kinetic</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">potential</a></li></ul></li> <li><a href="/wiki/Force" title="Force">Force</a></li> <li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li> <li><a href="/wiki/Impulse_(physics)" title="Impulse (physics)">Impulse</a></li> <li><span class="nowrap"><a href="/wiki/Inertia" title="Inertia">Inertia</a> / <a href="/wiki/Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></span></li> <li><a href="/wiki/Mass" title="Mass">Mass</a></li> <li><br /><a href="/wiki/Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Mechanical work</a></li> <li><br /><a href="/wiki/Moment_(physics)" title="Moment (physics)">Moment</a></li> <li><a href="/wiki/Momentum" title="Momentum">Momentum</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/Speed" title="Speed">Speed</a></li> <li><a href="/wiki/Time" title="Time">Time</a></li> <li><a href="/wiki/Torque" title="Torque">Torque</a></li> <li><a href="/wiki/Velocity" title="Velocity">Velocity</a></li> <li><a href="/wiki/Virtual_work" title="Virtual work">Virtual work</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"> <ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></b></div></li> <li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></b> <div class="plainlist"><ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="/wiki/Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="/wiki/Appell%27s_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Core topics</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Damping" title="Damping">Damping</a></li> <li><a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li> <li><a href="/wiki/Equations_of_motion" title="Equations of motion">Equations of motion</a></li> <li><a href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion"><span class="wrap">Euler's laws of motion</span></a></li> <li><a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a></li> <li><a href="/wiki/Friction" title="Friction">Friction</a></li> <li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul> </div> <ul><li><span class="nowrap"><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a> / <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></span></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Motion" title="Motion">Motion</a> (<a href="/wiki/Linear_motion" title="Linear motion">linear</a>)</li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation"><span class="wrap">Newton's law of universal gravitation</span></a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li> <li><a href="/wiki/Relative_velocity" title="Relative velocity">Relative velocity</a></li> <li><a href="/wiki/Rigid_body" title="Rigid body">Rigid body</a> <ul><li><a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li> <li><a href="/wiki/Euler%27s_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li> <li><a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li> <li><a href="/wiki/Vibration" title="Vibration">Vibration</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)"><a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Circular_motion" title="Circular motion">Circular motion</a></li> <li><a href="/wiki/Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li> <li><a href="/wiki/Centripetal_force" title="Centripetal force">Centripetal force</a></li> <li><a href="/wiki/Centrifugal_force" title="Centrifugal force">Centrifugal force</a> <ul><li><a href="/wiki/Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li> <li><a href="/wiki/Tangential_speed" title="Tangential speed">Tangential speed</a></li> <li><a href="/wiki/Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul> </div> <ul><li><a href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="/wiki/Angular_displacement" title="Angular displacement">displacement</a> / <a href="/wiki/Angular_frequency" title="Angular frequency">frequency</a> / <a href="/wiki/Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a></li> <li><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li> <li><a href="/wiki/Edmond_Halley" title="Edmond Halley">Halley</a></li> <li><a href="/wiki/Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li> <li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li> <li><a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">d'Alembert</a></li> <li><a href="/wiki/Alexis_Clairaut" title="Alexis Clairaut">Clairaut</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li> <li><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li> <li><a 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.navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classical_mechanics" title="Template:Classical mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classical_mechanics" title="Template talk:Classical mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics" title="Special:EditPage/Template:Classical mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Classical mechanics</b> is a <a href="/wiki/Theoretical_physics" title="Theoretical physics">physical theory</a> describing the <a href="/wiki/Motion" title="Motion">motion</a> of objects such as <a href="/wiki/Projectile" title="Projectile">projectiles</a>, parts of <a href="/wiki/Machine_(mechanical)" class="mw-redirect" title="Machine (mechanical)">machinery</a>, <a href="/wiki/Spacecraft" title="Spacecraft">spacecraft</a>, <a href="/wiki/Planets" class="mw-redirect" title="Planets">planets</a>, <a href="/wiki/Star" title="Star">stars</a>, and <a href="/wiki/Galaxies" class="mw-redirect" title="Galaxies">galaxies</a>. The development of classical mechanics involved <a href="/wiki/Scientific_Revolution" title="Scientific Revolution">substantial change in the methods and philosophy</a> of physics.<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> The qualifier <i>classical</i> distinguishes this type of mechanics from physics developed after the <a href="/wiki/History_of_physics#20th_century:_birth_of_modern_physics" title="History of physics">revolutions in physics of the early 20th century</a>, all of which revealed limitations in classical mechanics.<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> </p><p>The earliest formulation of classical mechanics is often referred to as <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a>. It consists of the physical concepts based on the 17th century foundational works of Sir <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>, and the mathematical methods invented by Newton, <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> and others to describe the motion of <a href="/wiki/Physical_body" class="mw-redirect" title="Physical body">bodies</a> under the influence of <a href="/wiki/Force" title="Force">forces</a>. Later, methods based on <a href="/wiki/Energy" title="Energy">energy</a> were developed by Euler, <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a>, <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> and others, leading to the development of <a href="/wiki/Analytical_mechanics" title="Analytical mechanics">analytical mechanics</a> (which includes <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a> and <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>). These advances, made predominantly in the 18th and 19th centuries, extended beyond earlier works; they are, with some modification, used in all areas of modern physics. </p><p>If the present state of an object that obeys the laws of classical mechanics is known, it is possible to <a href="/wiki/Determinism" title="Determinism">determine how it will move in the future</a>, and how it has moved in the past. <a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a> shows that the long term predictions of classical mechanics are not reliable. Classical mechanics provides accurate results when studying objects that are not extremely massive and have speeds not approaching the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. With objects about the size of an atom's diameter, it becomes necessary to use <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. To describe velocities approaching the speed of light, <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> is needed. In cases where objects become extremely massive, <a href="/wiki/General_relativity" title="General relativity">general relativity</a> becomes applicable. Some modern sources include <a href="/wiki/Relativistic_mechanics" title="Relativistic mechanics">relativistic mechanics</a> in classical physics, as representing the field in its most developed and accurate form. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Branches">Branches</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=1" title="Edit section: Branches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Traditional_division">Traditional division</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=2" title="Edit section: Traditional division"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Classical mechanics was traditionally divided into three main branches. <i><a href="/wiki/Statics" title="Statics">Statics</a></i> is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in <a href="/wiki/Mechanical_equilibrium" title="Mechanical equilibrium">equilibrium</a> with its environment.<sup id="cite_ref-Wright_3-0" class="reference"><a href="#cite_note-Wright-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <i><a href="/wiki/Kinematics" title="Kinematics">Kinematics</a></i> describes the <a href="/wiki/Motion_(physics)" class="mw-redirect" title="Motion (physics)">motion</a> of points, <a href="/wiki/Physical_object" title="Physical object">bodies</a> (objects), and systems of bodies (groups of objects) without considering the <a href="/wiki/Force" title="Force">forces</a> that cause them to move.<sup id="cite_ref-Whittaker_4-0" class="reference"><a href="#cite_note-Whittaker-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Beggs_5-0" class="reference"><a href="#cite_note-Beggs-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Wright_3-1" class="reference"><a href="#cite_note-Wright-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>.<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><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> <a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)"><i>Dynamics</i></a> goes beyond merely describing objects' behavior and also considers the forces which explain it. Some authors (for example, Taylor (2005)<sup id="cite_ref-Taylor_9-0" class="reference"><a href="#cite_note-Taylor-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and Greenwood (1997)<sup id="cite_ref-Greenwood_10-0" class="reference"><a href="#cite_note-Greenwood-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>) include <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> within classical dynamics. </p> <div class="mw-heading mw-heading3"><h3 id="Forces_vs._energy">Forces vs. energy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=3" title="Edit section: Forces vs. energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another division is based on the choice of mathematical formalism. Classical mechanics can be mathematically presented in multiple different ways. The physical content of these different formulations is the same, but they provide different insights and facilitate different types of calculations. While the term "Newtonian mechanics" is sometimes used as a synonym for non-relativistic classical physics, it can also refer to a particular formalism based on <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a>. Newtonian mechanics in this sense emphasizes force as a <a href="/wiki/Vector_(physics)" class="mw-redirect" title="Vector (physics)">vector</a> quantity.<sup id="cite_ref-Lanczos_11-0" class="reference"><a href="#cite_note-Lanczos-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>In contrast, <a href="/wiki/Analytical_mechanics" title="Analytical mechanics">analytical mechanics</a> uses <i><a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</a></i> properties of motion representing the system as a whole—usually its <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> and <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>. The <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a> are derived from the scalar quantity by some underlying principle about the scalar's <a href="/wiki/Calculus_of_variations" title="Calculus of variations">variation</a>. Two dominant branches of analytical mechanics are <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>, which uses generalized coordinates and corresponding generalized velocities in <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a>, and <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>, which uses coordinates and corresponding momenta in <a href="/wiki/Phase_space" title="Phase space">phase space</a>. Both formulations are equivalent by a <a href="/wiki/Legendre_transformation#Hamilton–Lagrange_mechanics" title="Legendre transformation">Legendre transformation</a> on the generalized coordinates, velocities and momenta; therefore, both contain the same information for describing the dynamics of a system. There are other formulations such as <a href="/wiki/Hamilton%E2%80%93Jacobi_theory" class="mw-redirect" title="Hamilton–Jacobi theory">Hamilton–Jacobi theory</a>, <a href="/wiki/Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a>, and <a href="/wiki/Appell%27s_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a>. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called the <a href="/wiki/Principle_of_least_action" class="mw-redirect" title="Principle of least action">principle of least action</a>. One result is <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a>, a statement which connects <a href="/wiki/Conservation_law" title="Conservation law">conservation laws</a> to their associated <a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">symmetries</a>. </p> <div class="mw-heading mw-heading3"><h3 id="By_region_of_application">By region of application</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=4" title="Edit section: By region of application"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alternatively, a division can be made by region of application: </p> <ul><li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial mechanics</a>, relating to <a href="/wiki/Star" title="Star">stars</a>, <a href="/wiki/Planet" title="Planet">planets</a> and other celestial bodies</li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a>, for materials modelled as a continuum, e.g., <a href="/wiki/Solid" title="Solid">solids</a> and <a href="/wiki/Fluid" title="Fluid">fluids</a> (i.e., <a href="/wiki/Liquid" title="Liquid">liquids</a> and <a href="/wiki/Gas" title="Gas">gases</a>).</li> <li><a href="/wiki/Relativistic_mechanics" title="Relativistic mechanics">Relativistic mechanics</a> (i.e. including the <a href="/wiki/Special_relativity" title="Special relativity">special</a> and <a href="/wiki/General_relativity" title="General relativity">general</a> theories of relativity), for bodies whose speed is close to the speed of light.</li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a>, which provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamic</a> properties of materials.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Description_of_objects_and_their_motion">Description of objects and their motion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=5" title="Edit section: Description of objects and their motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tir_parab%C3%B2lic.svg" class="mw-file-description"><img alt="diagram of parabolic projectile motion" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Tir_parab%C3%B2lic.svg/220px-Tir_parab%C3%B2lic.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Tir_parab%C3%B2lic.svg/330px-Tir_parab%C3%B2lic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Tir_parab%C3%B2lic.svg/440px-Tir_parab%C3%B2lic.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>The analysis of <a href="/wiki/Projectile_motion" title="Projectile motion">projectile motion</a> is a part of classical mechanics.</figcaption></figure> <p>For simplicity, classical mechanics often models real-world objects as <a href="/wiki/Point_particle" title="Point particle">point particles</a>, that is, objects with negligible size. The motion of a point particle is determined by a small number of <a href="/wiki/Parameter" title="Parameter">parameters</a>: its position, <a href="/wiki/Mass" title="Mass">mass</a>, and the <a href="/wiki/Force" title="Force">forces</a> applied to it. Classical mechanics also describes the more complex motions of extended non-pointlike objects. <a href="/wiki/Euler%27s_laws" class="mw-redirect" title="Euler's laws">Euler's laws</a> provide extensions to Newton's laws in this area. The concepts of <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> rely on the same <a href="/wiki/Calculus" title="Calculus">calculus</a> used to describe one-dimensional motion. The <a href="/wiki/Rocket_equation" class="mw-redirect" title="Rocket equation">rocket equation</a> extends the notion of rate of change of an object's momentum to include the effects of an object "losing mass". (These generalizations/extensions are derived from Newton's laws, say, by decomposing a solid body into a collection of points.) </p><p>In reality, the kind of objects that classical mechanics can describe always have a <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">non-zero</a> size. (The behavior of <i>very</i> small particles, such as the <a href="/wiki/Electron" title="Electron">electron</a>, is more accurately described by <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>.) Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a>, e.g., a <a href="/wiki/Baseball_(ball)" title="Baseball (ball)">baseball</a> can <a href="/wiki/Rotation" title="Rotation">spin</a> while it is moving. However, the results for point particles can be used to study such objects by treating them as <a href="https://en.wiktionary.org/wiki/composite" class="extiw" title="wikt:composite">composite</a> objects, made of a large number of collectively acting point particles. The <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> of a composite object behaves like a point particle. </p><p>Classical mechanics assumes that matter and energy have definite, knowable attributes such as location in space and speed. Non-relativistic mechanics also assumes that forces act instantaneously (see also <a href="/wiki/Action_at_a_distance" title="Action at a distance">Action at a distance</a>). </p> <div style="clear:right;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Kinematics">Kinematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=6" title="Edit section: Kinematics"><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/Kinematics" title="Kinematics">Kinematics</a></div> <table class="wikitable" style="float:right; margin:0 0 1em 1em;"> <caption>The <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> derived "mechanical"<br />(that is, not <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetic</a> or <a href="/wiki/Thermal_physics" title="Thermal physics">thermal</a>)<br />units with kg, m and <a href="/wiki/Second" title="Second">s</a> </caption> <tbody><tr> <td>position</td> <td>m </td></tr> <tr> <td>angular position/<a href="/wiki/Angle" title="Angle">angle</a></td> <td>unitless (radian) </td></tr> <tr> <td><a href="/wiki/Velocity" title="Velocity">velocity</a></td> <td>m·s<sup>−1</sup> </td></tr> <tr> <td><a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a></td> <td>s<sup>−1</sup> </td></tr> <tr> <td><a href="/wiki/Acceleration" title="Acceleration">acceleration</a></td> <td>m·s<sup>−2</sup> </td></tr> <tr> <td><a href="/wiki/Angular_acceleration" title="Angular acceleration">angular acceleration</a></td> <td>s<sup>−2</sup> </td></tr> <tr> <td><a href="/wiki/Jerk_(physics)" title="Jerk (physics)">jerk</a></td> <td>m·s<sup>−3</sup> </td></tr> <tr> <td>"angular jerk"</td> <td>s<sup>−3</sup> </td></tr> <tr> <td><a href="/wiki/Specific_energy" title="Specific energy">specific energy</a></td> <td>m<sup>2</sup>·s<sup>−2</sup> </td></tr> <tr> <td>absorbed dose rate</td> <td>m<sup>2</sup>·s<sup>−3</sup> </td></tr> <tr> <td><a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a></td> <td>kg·m<sup>2</sup> </td></tr> <tr> <td><a href="/wiki/Momentum" title="Momentum">momentum</a></td> <td>kg·m·s<sup>−1</sup> </td></tr> <tr> <td><a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a></td> <td>kg·m<sup>2</sup>·s<sup>−1</sup> </td></tr> <tr> <td><a href="/wiki/Force" title="Force">force</a></td> <td>kg·m·s<sup>−2</sup> </td></tr> <tr> <td><a href="/wiki/Torque" title="Torque">torque</a></td> <td>kg·m<sup>2</sup>·s<sup>−2</sup> </td></tr> <tr> <td><a href="/wiki/Energy" title="Energy">energy</a></td> <td>kg·m<sup>2</sup>·s<sup>−2</sup> </td></tr> <tr> <td><a href="/wiki/Power_(physics)" title="Power (physics)">power</a></td> <td>kg·m<sup>2</sup>·s<sup>−3</sup> </td></tr> <tr> <td><a href="/wiki/Pressure" title="Pressure">pressure</a> and <a href="/wiki/Energy_density" title="Energy density">energy density</a></td> <td>kg·m<sup>−1</sup>·s<sup>−2</sup> </td></tr> <tr> <td><a href="/wiki/Surface_tension" title="Surface tension">surface tension</a></td> <td>kg·s<sup>−2</sup> </td></tr> <tr> <td><a href="/wiki/Spring_constant" class="mw-redirect" title="Spring constant">spring constant</a></td> <td>kg·s<sup>−2</sup> </td></tr> <tr> <td><a href="/wiki/Irradiance" title="Irradiance">irradiance</a> and <a href="/wiki/Energy_flux" title="Energy flux">energy flux</a></td> <td>kg·s<sup>−3</sup> </td></tr> <tr> <td><a href="/wiki/Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a></td> <td>m<sup>2</sup>·s<sup>−1</sup> </td></tr> <tr> <td><a href="/wiki/Dynamic_viscosity" class="mw-redirect" title="Dynamic viscosity">dynamic viscosity</a></td> <td>kg·m<sup>−1</sup>·s<sup>−1</sup> </td></tr> <tr> <td><a href="/wiki/Density" title="Density">density</a> (mass density)</td> <td>kg·m<sup>−3</sup> </td></tr> <tr> <td><a href="/wiki/Specific_weight" title="Specific weight">specific weight</a> (weight density)</td> <td>kg·m<sup>−2</sup>·s<sup>−2</sup> </td></tr> <tr> <td><a href="/wiki/Number_density" title="Number density">number density</a></td> <td>m<sup>−3</sup> </td></tr> <tr> <td><a href="/wiki/Action_(physics)" title="Action (physics)">action</a></td> <td>kg·m<sup>2</sup>·s<sup>−1</sup> </td></tr></tbody></table> <p>The <i>position</i> of a <a href="/wiki/Point_particle" title="Point particle">point particle</a> is defined in relation to a <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> centered on an arbitrary fixed reference point in <a href="/wiki/Space" title="Space">space</a> called the origin <i>O</i>. A simple coordinate system might describe the position of a <a href="/wiki/Particle" title="Particle">particle</a> <i>P</i> with a <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vector</a> notated by an arrow labeled <b>r</b> that points from the origin <i>O</i> to point <i>P</i>. In general, the point particle does not need to be stationary relative to <i>O</i>. In cases where <i>P</i> is moving relative to <i>O</i>, <b>r</b> is defined as a function of <i>t</i>, <a href="/wiki/Time" title="Time">time</a>. In pre-Einstein relativity (known as <a href="/wiki/Galilean_relativity" class="mw-redirect" title="Galilean relativity">Galilean relativity</a>), time is considered an absolute, i.e., the <a href="/wiki/Time_interval" class="mw-redirect" title="Time interval">time interval</a> that is observed to elapse between any given pair of events is the same for all observers.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> In addition to relying on <a href="/wiki/Absolute_time" class="mw-redirect" title="Absolute time">absolute time</a>, classical mechanics assumes <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> for the structure of space.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Velocity_and_speed">Velocity and speed</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=7" title="Edit section: Velocity and speed"><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 articles: <a href="/wiki/Velocity" title="Velocity">Velocity</a> and <a href="/wiki/Speed" title="Speed">speed</a></div> <p>The <i><a href="/wiki/Velocity" title="Velocity">velocity</a></i>, or the <a href="/wiki/Calculus" title="Calculus">rate of change</a> of displacement with time, is defined as the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the position with respect to time: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} ={\mathrm {d} \mathbf {r} \over \mathrm {d} t}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\mathrm {d} \mathbf {r} \over \mathrm {d} t}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92a7d2f19f00b2972ba90635da426701bbd94bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:8.127ex; height:5.509ex;" alt="{\displaystyle \mathbf {v} ={\mathrm {d} \mathbf {r} \over \mathrm {d} t}\,\!}"></span>.</dd></dl> <p>In classical mechanics, velocities are directly additive and subtractive. For example, if one car travels east at 60 km/h and passes another car traveling in the same direction at 50 km/h, the slower car perceives the faster car as traveling east at <span class="nowrap">60 − 50 = 10 km/h</span>. However, from the perspective of the faster car, the slower car is moving 10 km/h to the west, often denoted as −10 km/h where the sign implies opposite direction. Velocities are directly additive as <a href="https://en.wikibooks.org/wiki/Physics_with_Calculus/Mechanics/Scalar_and_Vector_Quantities" class="extiw" title="b:Physics with Calculus/Mechanics/Scalar and Vector Quantities">vector quantities</a>; they must be dealt with using <a href="/wiki/Vector_analysis" class="mw-redirect" title="Vector analysis">vector analysis</a>. </p><p>Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector <span class="nowrap"><b>u</b> = <i>u</i><b>d</b></span> and the velocity of the second object by the vector <span class="nowrap"><b>v</b> = <i>v</i><b>e</b></span>, where <i>u</i> is the speed of the first object, <i>v</i> is the speed of the second object, and <b>d</b> and <b>e</b> are <a href="/wiki/Unit_vector" title="Unit vector">unit vectors</a> in the directions of motion of each object respectively, then the velocity of the first object as seen by the second object is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} '=\mathbf {u} -\mathbf {v} \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} '=\mathbf {u} -\mathbf {v} \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc94c55ea3e6a065dc748237bd10b4d8f07622e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.039ex; height:2.676ex;" alt="{\displaystyle \mathbf {u} '=\mathbf {u} -\mathbf {v} \,.}"></span></dd></dl> <p>Similarly, the first object sees the velocity of the second object as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v'} =\mathbf {v} -\mathbf {u} \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">v</mi> <mo>′</mo> </msup> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v'} =\mathbf {v} -\mathbf {u} \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bb53f1542de511363cac3689c5f8428b17eb614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.965ex; height:2.676ex;" alt="{\displaystyle \mathbf {v'} =\mathbf {v} -\mathbf {u} \,.}"></span></dd></dl> <p>When both objects are moving in the same direction, this equation can be simplified to: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} '=(u-v)\mathbf {d} \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} '=(u-v)\mathbf {d} \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1624c018b5557d9536c0744f65cfc4dadaea77d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.895ex; height:3.009ex;" alt="{\displaystyle \mathbf {u} '=(u-v)\mathbf {d} \,.}"></span></dd></dl> <p>Or, by ignoring direction, the difference can be given in terms of speed only: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u'=u-v\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>u</mi> <mo>−</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u'=u-v\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8018d62fee8045158d13ba81e7e1effe8c4d2d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.444ex; height:2.676ex;" alt="{\displaystyle u'=u-v\,.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Acceleration">Acceleration</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=8" title="Edit section: Acceleration"><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/Acceleration" title="Acceleration">Acceleration</a></div> <p>The <i><a href="/wiki/Acceleration" title="Acceleration">acceleration</a></i>, or rate of change of velocity, is the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the velocity with respect to time (the <a href="/wiki/Derivative" title="Derivative">second derivative</a> of the position with respect to time): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ={\mathrm {d} \mathbf {v} \over \mathrm {d} t}={\mathrm {d^{2}} \mathbf {r} \over \mathrm {d} t^{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ={\mathrm {d} \mathbf {v} \over \mathrm {d} t}={\mathrm {d^{2}} \mathbf {r} \over \mathrm {d} t^{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d60f14ca073ab34f4e95288db447b05a9b8f7311" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.968ex; height:6.009ex;" alt="{\displaystyle \mathbf {a} ={\mathrm {d} \mathbf {v} \over \mathrm {d} t}={\mathrm {d^{2}} \mathbf {r} \over \mathrm {d} t^{2}}.}"></span></dd></dl> <p>Acceleration represents the velocity's change over time. Velocity can change in magnitude, direction, or both. Occasionally, a decrease in the magnitude of velocity "<i>v</i>" is referred to as <i>deceleration</i>, but generally any change in the velocity over time, including deceleration, is referred to as acceleration. </p> <div class="mw-heading mw-heading4"><h4 id="Frames_of_reference">Frames of reference</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=9" title="Edit section: Frames of reference"><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 articles: <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a> and <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a></div> <p>While the position, velocity and acceleration of a <a href="/wiki/Particle" title="Particle">particle</a> can be described with respect to any <a href="/wiki/Observer_(special_relativity)" title="Observer (special relativity)">observer</a> in any state of motion, classical mechanics assumes the existence of a special family of <a href="/wiki/Frame_of_reference" title="Frame of reference">reference frames</a> in which the mechanical laws of nature take a comparatively simple form. These special reference frames are called <a href="/wiki/Inertial_frames" class="mw-redirect" title="Inertial frames">inertial frames</a>. An inertial frame is an idealized frame of reference within which an object with zero net force acting upon it moves with a constant velocity; that is, it is either at rest or moving uniformly in a straight line. In an inertial frame Newton's law of motion, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=ma}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>m</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=ma}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca4e42b7d6d66f52294364928cb5f7c590f514c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.109ex; height:2.176ex;" alt="{\displaystyle F=ma}"></span>, is valid.<sup id="cite_ref-Goldstein_14-0" class="reference"><a href="#cite_note-Goldstein-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 185">: 185 </span></sup> </p><p><a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frames</a> accelerate in relation to another inertial frame. A body rotating with respect to an inertial frame is not an inertial frame.<sup id="cite_ref-Goldstein_14-1" class="reference"><a href="#cite_note-Goldstein-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> When viewed from an inertial frame, particles in the non-inertial frame appear to move in ways not explained by forces from existing fields in the reference frame. Hence, it appears that there are other forces that enter the equations of motion solely as a result of the relative acceleration. These forces are referred to as <a href="/wiki/Fictitious_force" title="Fictitious force">fictitious forces</a>, inertia forces, or pseudo-forces. </p><p>Consider two <a href="/wiki/Reference_frames" class="mw-redirect" title="Reference frames">reference frames</a> <i>S</i> and <var>S'</var>. For observers in each of the reference frames an event has space-time coordinates of (<i>x</i>,<i>y</i>,<i>z</i>,<i>t</i>) in frame <i>S</i> and (<var>x'</var>,<var>y'</var>,<var>z'</var>,<var>t'</var>) in frame <var>S'</var>. Assuming time is measured the same in all reference frames, if we require <span class="nowrap"><i>x</i> = <var>x'</var></span> when <span class="nowrap"><i>t</i> = 0</span>, then the relation between the space-time coordinates of the same event observed from the reference frames <var>S'</var> and <i>S</i>, which are moving at a relative velocity <i>u</i> in the <i>x</i> direction, is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x'&=x-tu,\\y'&=y,\\z'&=z,\\t'&=t.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>t</mi> <mi>u</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x'&=x-tu,\\y'&=y,\\z'&=z,\\t'&=t.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86e7c53d56693e48081903d4ca1085b88b88fb54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:12.851ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}x'&=x-tu,\\y'&=y,\\z'&=z,\\t'&=t.\end{aligned}}}"></span></dd></dl> <p>This set of formulas defines a <a href="/wiki/Group_transformation" class="mw-redirect" title="Group transformation">group transformation</a> known as the <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a> (informally, the <i>Galilean transform</i>). This group is a limiting case of the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> used in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. The limiting case applies when the velocity <i>u</i> is very small compared to <i>c</i>, the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. </p><p>The transformations have the following consequences: </p> <ul><li><b>v</b>′ = <b>v</b> − <b>u</b> (the velocity <b>v</b>′ of a particle from the perspective of <i>S</i>′ is slower by <b>u</b> than its velocity <b>v</b> from the perspective of <i>S</i>)</li> <li><b>a</b>′ = <b>a</b> (the acceleration of a particle is the same in any inertial reference frame)</li> <li><b>F</b>′ = <b>F</b> (the force on a particle is the same in any inertial reference frame)</li> <li>the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> is not a constant in classical mechanics, nor does the special position given to the speed of light in <a href="/wiki/Relativistic_mechanics" title="Relativistic mechanics">relativistic mechanics</a> have a counterpart in classical mechanics.</li></ul> <p>For some problems, it is convenient to use rotating coordinates (reference frames). Thereby one can either keep a mapping to a convenient inertial frame, or introduce additionally a fictitious <a href="/wiki/Centrifugal_force_(fictitious)" class="mw-redirect" title="Centrifugal force (fictitious)">centrifugal force</a> and <a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Newtonian_mechanics">Newtonian mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=10" title="Edit section: Newtonian mechanics"><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 articles: <a href="/wiki/Force" title="Force">Force</a> and <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></div> <p>A force in physics is any action that causes an object's velocity to change; that is, to accelerate. A force originates from within a <a href="/wiki/Field_(physics)" title="Field (physics)">field</a>, such as an electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), or gravitational field (caused by mass), among others. </p><p><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> was the first to mathematically express the relationship between <a href="/wiki/Force" title="Force">force</a> and <a href="/wiki/Momentum" title="Momentum">momentum</a>. Some physicists interpret <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law of motion</a> as a definition of force and mass, while others consider it a fundamental postulate, a law of nature.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law": </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ={\mathrm {d} \mathbf {p} \over \mathrm {d} t}={\mathrm {d} (m\mathbf {v} ) \over \mathrm {d} t}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ={\mathrm {d} \mathbf {p} \over \mathrm {d} t}={\mathrm {d} (m\mathbf {v} ) \over \mathrm {d} t}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/778c72a4b2dece458119de6b9b52e67abfd59e39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.53ex; height:5.843ex;" alt="{\displaystyle \mathbf {F} ={\mathrm {d} \mathbf {p} \over \mathrm {d} t}={\mathrm {d} (m\mathbf {v} ) \over \mathrm {d} t}.}"></span></dd></dl> <p>The quantity <i>m</i><b>v</b> is called the (<a href="/wiki/Canonical_momentum" class="mw-redirect" title="Canonical momentum">canonical</a>) <a href="/wiki/Momentum" title="Momentum">momentum</a>. The net force on a particle is thus equal to the rate of change of the momentum of the particle with time. Since the definition of acceleration is <span class="nowrap"><b>a</b> = d<b>v</b>/d<i>t</i></span>, the second law can be written in the simplified and more familiar form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =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> <mspace width="thinmathspace" /> <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/f8dc269e16348cc89ff75d9516ea7cfba7731e5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.155ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} =m\mathbf {a} \,.}"></span></dd></dl> <p>So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a>, which is called the <i>equation of motion</i>. </p><p>As an example, assume that friction is the only force acting on the particle, and that it may be modeled as a function of the velocity of the particle, for example: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{\rm {R}}=-\lambda \mathbf {v} \,,}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\rm {R}}=-\lambda \mathbf {v} \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca3bb72778add17acc29fa26fa28e491949cfd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.831ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{\rm {R}}=-\lambda \mathbf {v} \,,}"></span></dd></dl> <p>where <i>λ</i> is a positive constant, the negative sign states that the force is opposite the sense of the velocity. Then the equation of motion is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\lambda \mathbf {v} =m\mathbf {a} =m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\lambda \mathbf {v} =m\mathbf {a} =m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7073fe3f53a72a97348c35f68007ddd0c5e003fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.725ex; height:5.509ex;" alt="{\displaystyle -\lambda \mathbf {v} =m\mathbf {a} =m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}\,.}"></span></dd></dl> <p>This can be <a href="/wiki/Antiderivative" title="Antiderivative">integrated</a> to obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =\mathbf {v} _{0}e^{{-\lambda t}/{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =\mathbf {v} _{0}e^{{-\lambda t}/{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/956862d2f1f16e86e6e1cbc603d968ce8101ae1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.386ex; height:3.176ex;" alt="{\displaystyle \mathbf {v} =\mathbf {v} _{0}e^{{-\lambda t}/{m}}}"></span></dd></dl> <p>where <b>v</b><sub>0</sub> is the initial velocity. This means that the velocity of this particle <a href="/wiki/Exponential_decay" title="Exponential decay">decays exponentially</a> to zero as time progresses. In this case, an equivalent viewpoint is that the kinetic energy of the particle is absorbed by friction (which converts it to heat energy in accordance with the <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a>), and the particle is slowing down. This expression can be further integrated to obtain the position <b>r</b> of the particle as a function of time. </p><p>Important forces include the <a href="/wiki/Gravity" title="Gravity">gravitational force</a> and the <a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a> for <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>. In addition, <a href="/wiki/Newton%27s_third_law" class="mw-redirect" title="Newton's third law">Newton's third law</a> can sometimes be used to deduce the forces acting on a particle: if it is known that particle <i>A</i> exerts a force <b>F</b> on another particle <i>B</i>, it follows that <i>B</i> must exert an equal and opposite <i>reaction force</i>, −<b>F</b>, on <i>A</i>. The strong form of Newton's third law requires that <b>F</b> and −<b>F</b> act along the line connecting <i>A</i> and <i>B</i>, while the weak form does not. Illustrations of the weak form of Newton's third law are often found for magnetic forces.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (January 2016)">clarification needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Work_and_energy">Work and energy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=11" title="Edit section: Work and energy"><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 articles: <a href="/wiki/Work_(physics)" title="Work (physics)">Work (physics)</a>, <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a>, and <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a></div> <p>If a constant force <b>F</b> is applied to a particle that makes a displacement Δ<b>r</b>,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> the <i>work done</i> by the force is defined as the <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a> of the force and displacement vectors: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\mathbf {F} \cdot \Delta \mathbf {r} \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\mathbf {F} \cdot \Delta \mathbf {r} \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1c00fa8d1dc4ff11544e8a53580eea0e3140460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.967ex; height:2.176ex;" alt="{\displaystyle W=\mathbf {F} \cdot \Delta \mathbf {r} \,.}"></span></dd></dl> <p>More generally, if the force varies as a function of position as the particle moves from <b>r</b><sub>1</sub> to <b>r</b><sub>2</sub> along a path <i>C</i>, the work done on the particle is given by the <a href="/wiki/Line_integral" title="Line integral">line integral</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{C}\mathbf {F} (\mathbf {r} )\cdot \mathrm {d} \mathbf {r} \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</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="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}\mathbf {F} (\mathbf {r} )\cdot \mathrm {d} \mathbf {r} \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e348ec36240e84e6e941288fc9e4456321c67857" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.396ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}\mathbf {F} (\mathbf {r} )\cdot \mathrm {d} \mathbf {r} \,.}"></span></dd></dl> <p>If the work done in moving the particle from <b>r</b><sub>1</sub> to <b>r</b><sub>2</sub> is the same no matter what path is taken, the force is said to be <a href="/wiki/Conservative_force" title="Conservative force">conservative</a>. <a href="/wiki/Gravity" title="Gravity">Gravity</a> is a conservative force, as is the force due to an idealized <a href="/wiki/Spring_(device)" title="Spring (device)">spring</a>, as given by <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>. The force due to <a href="/wiki/Friction" title="Friction">friction</a> is non-conservative. </p><p>The <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> <i>E</i><sub>k</sub> of a particle of mass <i>m</i> travelling at speed <i>v</i> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {k} }={\tfrac {1}{2}}mv^{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {k} }={\tfrac {1}{2}}mv^{2}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37fdbdcc4d6eecf5020be8b1b89501455448b0f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.828ex; height:3.509ex;" alt="{\displaystyle E_{\mathrm {k} }={\tfrac {1}{2}}mv^{2}\,.}"></span></dd></dl> <p>For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles. </p><p>The <a href="/wiki/Work%E2%80%93energy_theorem" class="mw-redirect" title="Work–energy theorem">work–energy theorem</a> states that for a particle of constant mass <i>m</i>, the total work <i>W</i> done on the particle as it moves from position <b>r</b><sub>1</sub> to <b>r</b><sub>2</sub> is equal to the change in <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> <i>E</i><sub>k</sub> of the particle: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\Delta E_{\mathrm {k} }=E_{\mathrm {k_{2}} }-E_{\mathrm {k_{1}} }={\tfrac {1}{2}}m\left(v_{2}^{\,2}-v_{1}^{\,2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mrow> </msub> <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> <mo>(</mo> <mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\Delta E_{\mathrm {k} }=E_{\mathrm {k_{2}} }-E_{\mathrm {k_{1}} }={\tfrac {1}{2}}m\left(v_{2}^{\,2}-v_{1}^{\,2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d80ed5cee5b41df1249b7c27b4a3f28973d853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.844ex; height:3.509ex;" alt="{\displaystyle W=\Delta E_{\mathrm {k} }=E_{\mathrm {k_{2}} }-E_{\mathrm {k_{1}} }={\tfrac {1}{2}}m\left(v_{2}^{\,2}-v_{1}^{\,2}\right).}"></span></dd></dl> <p>Conservative forces can be expressed as the <a href="/wiki/Gradient" title="Gradient">gradient</a> of a scalar function, known as the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> and denoted <i>E</i><sub>p</sub>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =-\mathbf {\nabla } E_{\mathrm {p} }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =-\mathbf {\nabla } E_{\mathrm {p} }\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a51cf0584c83189ac2f83d9ab41bec28dfb410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.421ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} =-\mathbf {\nabla } E_{\mathrm {p} }\,.}"></span></dd></dl> <p>If all the forces acting on a particle are conservative, and <i>E</i><sub>p</sub> is the total potential energy (which is defined as a work of involved forces to rearrange mutual positions of bodies), obtained by summing the potential energies corresponding to each force </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} \cdot \Delta \mathbf {r} =-\mathbf {\nabla } E_{\mathrm {p} }\cdot \Delta \mathbf {r} =-\Delta E_{\mathrm {p} }\,.}"> <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 mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} \cdot \Delta \mathbf {r} =-\mathbf {\nabla } E_{\mathrm {p} }\cdot \Delta \mathbf {r} =-\Delta E_{\mathrm {p} }\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36522d278cbaaed0dbffb3cf2f332589076e65b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.559ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} \cdot \Delta \mathbf {r} =-\mathbf {\nabla } E_{\mathrm {p} }\cdot \Delta \mathbf {r} =-\Delta E_{\mathrm {p} }\,.}"></span></dd></dl> <p>The decrease in the potential energy is equal to the increase in the kinetic energy </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\Delta E_{\mathrm {p} }=\Delta E_{\mathrm {k} }\Rightarrow \Delta (E_{\mathrm {k} }+E_{\mathrm {p} })=0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> <mo stretchy="false">⇒</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\Delta E_{\mathrm {p} }=\Delta E_{\mathrm {k} }\Rightarrow \Delta (E_{\mathrm {k} }+E_{\mathrm {p} })=0\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc2b29443d2a7bef0e623a0cb59a25e038b3ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.627ex; height:3.009ex;" alt="{\displaystyle -\Delta E_{\mathrm {p} }=\Delta E_{\mathrm {k} }\Rightarrow \Delta (E_{\mathrm {k} }+E_{\mathrm {p} })=0\,.}"></span></dd></dl> <p>This result is known as <i>conservation of energy</i> and states that the total <a href="/wiki/Energy" title="Energy">energy</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum E=E_{\mathrm {k} }+E_{\mathrm {p} }\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum E=E_{\mathrm {k} }+E_{\mathrm {p} }\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce2a847aec2c86d4e6419bd982adbc666758c02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.167ex; height:3.843ex;" alt="{\displaystyle \sum E=E_{\mathrm {k} }+E_{\mathrm {p} }\,,}"></span></dd></dl> <p>is constant in time. It is often useful, because many commonly encountered forces are conservative. </p> <div class="mw-heading mw-heading2"><h2 id="Lagrangian_mechanics">Lagrangian mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=12" title="Edit section: Lagrangian mechanics"><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/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></div> <p><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a> is a formulation of classical mechanics founded on the <a href="/wiki/Stationary-action_principle" class="mw-redirect" title="Stationary-action principle">stationary-action principle</a> (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> in his presentation to the Turin Academy of Science in 1760<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> culminating in his 1788 grand opus, <i><a href="/wiki/M%C3%A9canique_analytique" title="Mécanique analytique">Mécanique analytique</a></i>. Lagrangian mechanics describes a mechanical system as a pair <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 (M,L)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (M,L)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc409055f0960592f2dc1e3d80263fd62af469f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.868ex; height:2.843ex;" alt="{\textstyle (M,L)}"></span> consisting of a <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a> <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/913ace920108f7552777e36ac0b7ee3f5093a088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\textstyle M}"></span> and a smooth function <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb88de7e4d31737dae8f02575033272f29e6720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\textstyle L}"></span> within that space called a Lagrangian. For many systems, <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 L=T-V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>T</mi> <mo>−</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle L=T-V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc3468e5ecce3f1b9eb4d80d76ce04bcf70421b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.592ex; height:2.509ex;" alt="{\textstyle L=T-V,}"></span> 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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db299e88e5485f250f4ba15530469c8c6080a8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\textstyle T}"></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="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> are the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic</a> and <a href="/wiki/Potential_energy" title="Potential energy">potential</a> energy of the system, respectively. The stationary action principle requires that the <a href="/wiki/Action_(physics)#Action_(functional)" title="Action (physics)">action functional</a> of the system derived from <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb88de7e4d31737dae8f02575033272f29e6720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\textstyle L}"></span> must remain at a stationary point (a <a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">maximum</a>, <a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">minimum</a>, or <a href="/wiki/Saddle_point" title="Saddle point">saddle</a>) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Hamiltonian_mechanics">Hamiltonian mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=13" title="Edit section: Hamiltonian mechanics"><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/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></div> <p><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a> emerged in 1833 as a reformulation of <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>. Introduced by <a href="/wiki/Sir_William_Rowan_Hamilton" class="mw-redirect" title="Sir William Rowan Hamilton">Sir William Rowan Hamilton</a>,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Hamiltonian mechanics replaces (generalized) velocities <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 {\dot {q}}^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {q}}^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27052baf4d640bd44a642d1f450ea3819a2b4ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.177ex; height:3.009ex;" alt="{\displaystyle {\dot {q}}^{i}}"></span> used in Lagrangian mechanics with (generalized) <i>momenta</i>. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, <a href="/wiki/Symplectic_geometry" title="Symplectic geometry">symplectic geometry</a> and <a href="/wiki/Poisson_structure" class="mw-redirect" title="Poisson structure">Poisson structures</a>) and serves as a link between classical and <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. </p><p>In this formalism, the dynamics of a system are governed by Hamilton's equations, which express the time derivatives of position and momentum variables in terms of <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> of a function called the Hamiltonian: <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 {\mathrm {d} {\boldsymbol {q}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}},\quad {\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">q</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">p</mi> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">p</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">q</mi> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} {\boldsymbol {q}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}},\quad {\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64a55e53153f6ed6319f1f8c388e88fcf313dd2a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.177ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} {\boldsymbol {q}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}},\quad {\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}.}"></span> The Hamiltonian is the <a href="/wiki/Legendre_transform" class="mw-redirect" title="Legendre transform">Legendre transform</a> of the Lagrangian, and in many situations of physical interest it is equal to the total energy of the system. </p> <div class="mw-heading mw-heading2"><h2 id="Limits_of_validity">Limits of validity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=14" title="Edit section: Limits of validity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Modernphysicsfields_2.svg" class="mw-file-description"><img alt="two by two chart of mechanics for size by speed" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Modernphysicsfields_2.svg/390px-Modernphysicsfields_2.svg.png" decoding="async" width="390" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Modernphysicsfields_2.svg/585px-Modernphysicsfields_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Modernphysicsfields_2.svg/780px-Modernphysicsfields_2.svg.png 2x" data-file-width="787" data-file-height="355" /></a><figcaption>Domain of validity for classical mechanics</figcaption></figure> <p>Many branches of classical mechanics are simplifications or approximations of more accurate forms; two of the most accurate being <a href="/wiki/General_relativity" title="General relativity">general relativity</a> and relativistic <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>. <a href="/wiki/Geometric_optics" class="mw-redirect" title="Geometric optics">Geometric optics</a> is an approximation to the <a href="/wiki/Quantum_optics" title="Quantum optics">quantum theory of light</a>, and does not have a superior "classical" form. </p><p>When both quantum mechanics and classical mechanics cannot apply, such as at the quantum level with many degrees of freedom, <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> (QFT) is of use. QFT deals with small distances, and large speeds with many degrees of freedom as well as the possibility of any change in the number of particles throughout the interaction. When treating large degrees of freedom at the macroscopic level, <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> becomes useful. Statistical mechanics describes the behavior of large (but countable) numbers of particles and their interactions as a whole at the macroscopic level. Statistical mechanics is mainly used in <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> for systems that lie outside the bounds of the assumptions of classical thermodynamics. In the case of high <a href="/wiki/Velocity" title="Velocity">velocity</a> objects approaching the speed of light, classical mechanics is enhanced by <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. In case that objects become extremely heavy (i.e., their <a href="/wiki/Schwarzschild_radius" title="Schwarzschild radius">Schwarzschild radius</a> is not negligibly small for a given application), deviations from <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a> become apparent and can be quantified by using the <a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">parameterized post-Newtonian formalism</a>. In that case, <a href="/wiki/General_relativity" title="General relativity">general relativity</a> (GR) becomes applicable. However, until now there is no theory of <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a> unifying GR and QFT in the sense that it could be used when objects become extremely small and heavy.<sup><a class="mw-selflink-fragment" href="#cite_note-Hamber_2009-4">[4]</a><a class="mw-selflink-fragment" href="#cite_note-5">[5]</a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Newtonian_approximation_to_special_relativity">Newtonian approximation to special relativity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=15" title="Edit section: Newtonian approximation to special relativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In special relativity, the momentum of a particle is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} ={\frac {m\mathbf {v} }{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} ={\frac {m\mathbf {v} }{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad7ef6e1590c4dd00c12d2cde7168fd29fe439a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:15.245ex; height:7.509ex;" alt="{\displaystyle \mathbf {p} ={\frac {m\mathbf {v} }{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,,}"></span></dd></dl> <p>where <i>m</i> is the particle's rest mass, <b>v</b> its velocity, <i>v</i> is the modulus of <b>v</b>, and <i>c</i> is the speed of light. </p><p>If <i>v</i> is very small compared to <i>c</i>, <i>v</i><sup>2</sup>/<i>c</i><sup>2</sup> is approximately zero, and so </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} \approx m\mathbf {v} \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>≈</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} \approx m\mathbf {v} \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82fe45bbd6b2c8be946c98a01df3b5588b4e1ad8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.069ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} \approx m\mathbf {v} \,.}"></span></dd></dl> <p>Thus the Newtonian equation <span class="nowrap"><b>p</b> = <i>m</i><b>v</b></span> is an approximation of the relativistic equation for bodies moving with low speeds compared to the speed of light. </p><p>For example, the relativistic cyclotron frequency of a <a href="/wiki/Cyclotron" title="Cyclotron">cyclotron</a>, <a href="/wiki/Gyrotron" title="Gyrotron">gyrotron</a>, or high voltage <a href="/wiki/Magnetron" class="mw-redirect" title="Magnetron">magnetron</a> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=f_{\mathrm {c} }{\frac {m_{0}}{m_{0}+{\frac {T}{c^{2}}}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=f_{\mathrm {c} }{\frac {m_{0}}{m_{0}+{\frac {T}{c^{2}}}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1488fc3c02e8e4ecea7e9c71ec8c77c4f420c604" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:16.663ex; height:6.509ex;" alt="{\displaystyle f=f_{\mathrm {c} }{\frac {m_{0}}{m_{0}+{\frac {T}{c^{2}}}}}\,,}"></span></dd></dl> <p>where <i>f</i><sub>c</sub> is the classical frequency of an electron (or other charged particle) with kinetic energy <i>T</i> and (<a href="/wiki/Invariant_mass" title="Invariant mass">rest</a>) mass <i>m</i><sub>0</sub> circling in a magnetic field. The (rest) mass of an electron is 511 keV. So the frequency correction is 1% for a magnetic vacuum tube with a 5.11 kV direct current accelerating voltage. </p> <div class="mw-heading mw-heading3"><h3 id="Classical_approximation_to_quantum_mechanics">Classical approximation to quantum mechanics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=16" title="Edit section: Classical approximation to quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The ray approximation of classical mechanics breaks down when the <a href="/wiki/De_Broglie_wavelength" class="mw-redirect" title="De Broglie wavelength">de Broglie wavelength</a> is not much smaller than other dimensions of the system. For non-relativistic particles, this wavelength is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {h}{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>p</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {h}{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba64e374ccd3f05ca8b646070a27e94a2b28921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.629ex; height:5.843ex;" alt="{\displaystyle \lambda ={\frac {h}{p}}}"></span></dd></dl> <p>where <i>h</i> is the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a> and <i>p</i> is the momentum. </p><p>Again, this happens with <a href="/wiki/Electrons" class="mw-redirect" title="Electrons">electrons</a> before it happens with heavier particles. For example, the electrons used by <a href="/wiki/Clinton_Davisson" title="Clinton Davisson">Clinton Davisson</a> and <a href="/wiki/Lester_Germer" title="Lester Germer">Lester Germer</a> in 1927, accelerated by 54 V, had a wavelength of 0.167 nm, which was long enough to exhibit a single <a href="/wiki/Diffraction" title="Diffraction">diffraction</a> <a href="/wiki/Side_lobe" class="mw-redirect" title="Side lobe">side lobe</a> when reflecting from the face of a nickel <a href="/wiki/Crystal" title="Crystal">crystal</a> with atomic spacing of 0.215 nm. With a larger <a href="/wiki/Vacuum_chamber" title="Vacuum chamber">vacuum chamber</a>, it would seem relatively easy to increase the <a href="/wiki/Angular_resolution" title="Angular resolution">angular resolution</a> from around a radian to a <a href="/wiki/Milliradian" title="Milliradian">milliradian</a> and see quantum diffraction from the periodic patterns of <a href="/wiki/Integrated_circuit" title="Integrated circuit">integrated circuit</a> computer memory. </p><p>More practical examples of the failure of classical mechanics on an engineering scale are conduction by <a href="/wiki/Quantum_tunneling" class="mw-redirect" title="Quantum tunneling">quantum tunneling</a> in <a href="/wiki/Tunnel_diode" title="Tunnel diode">tunnel diodes</a> and very narrow <a href="/wiki/Transistor" title="Transistor">transistor</a> <a href="/wiki/Gate_(transistor)" class="mw-redirect" title="Gate (transistor)">gates</a> in <a href="/wiki/Integrated_circuit" title="Integrated circuit">integrated circuits</a>. </p><p>Classical mechanics is the same extreme <a href="/wiki/High_frequency_approximation" class="mw-redirect" title="High frequency approximation">high frequency approximation</a> as <a href="/wiki/Geometric_optics" class="mw-redirect" title="Geometric optics">geometric optics</a>. It is more often accurate because it describes particles and bodies with <a href="/wiki/Rest_mass" class="mw-redirect" title="Rest mass">rest mass</a>. These have more momentum and therefore shorter De Broglie wavelengths than massless particles, such as light, with the same kinetic energies. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=17" title="Edit section: History"><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/History_of_classical_mechanics" title="History of classical mechanics">History of classical mechanics</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For a chronological guide, see <a href="/wiki/Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline of classical mechanics</a>.</div> <p>The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in <a href="/wiki/Science" title="Science">science</a>, <a href="/wiki/Engineering" title="Engineering">engineering</a>, and <a href="/wiki/Technology" title="Technology">technology</a>. The development of classical mechanics lead to the development of many areas of mathematics.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 54">: 54 </span></sup> </p><p>Some <a href="/wiki/Greek_philosophers" class="mw-redirect" title="Greek philosophers">Greek philosophers</a> of antiquity, among them <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, founder of <a href="/wiki/Aristotelian_physics" title="Aristotelian physics">Aristotelian physics</a>, may have been the first to maintain the idea that "everything happens for a reason" and that theoretical principles can assist in the understanding of nature. While to a modern reader, many of these preserved ideas come forth as eminently reasonable, there is a conspicuous lack of both mathematical <a href="/wiki/Theory" title="Theory">theory</a> and controlled <a href="/wiki/Experiment" title="Experiment">experiment</a>, as we know it. These later became decisive factors in forming modern science, and their early application came to be known as classical mechanics. In his <i>Elementa super demonstrationem ponderum</i>, medieval mathematician <a href="/wiki/Jordanus_de_Nemore" title="Jordanus de Nemore">Jordanus de Nemore</a> introduced the concept of "positional <a href="/wiki/Gravity" title="Gravity">gravity</a>" and the use of component <a href="/wiki/Forces" class="mw-redirect" title="Forces">forces</a>. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Theory_of_impetus.svg" class="mw-file-description"><img alt="a diagram of Theory of impetus of Albert of Saxony with a b c d" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Theory_of_impetus.svg/180px-Theory_of_impetus.svg.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Theory_of_impetus.svg/270px-Theory_of_impetus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Theory_of_impetus.svg/360px-Theory_of_impetus.svg.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>Three stage <a href="/wiki/Theory_of_impetus" title="Theory of impetus">Theory of impetus</a> according to <a href="/wiki/Albert_of_Saxony_(philosopher)" title="Albert of Saxony (philosopher)">Albert of Saxony</a></figcaption></figure> <p>The first published <a href="/wiki/Causal" class="mw-redirect" title="Causal">causal</a> explanation of the motions of <a href="/wiki/Planets" class="mw-redirect" title="Planets">planets</a> was Johannes Kepler's <i><a href="/wiki/Astronomia_nova" title="Astronomia nova">Astronomia nova</a>,</i> published in 1609. He concluded, based on <a href="/wiki/Tycho_Brahe" title="Tycho Brahe">Tycho Brahe</a>'s observations on the orbit of <a href="/wiki/Mars" title="Mars">Mars</a>, that the planet's orbits were <a href="/wiki/Ellipse" title="Ellipse">ellipses</a>. This break with <a href="/wiki/Ancient_philosophy" title="Ancient philosophy">ancient thought</a> was happening around the same time that <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a> was proposing abstract mathematical laws for the motion of objects. He may (or may not) have performed the famous experiment of dropping two cannonballs of different weights from the <a href="/wiki/Leaning_Tower_of_Pisa" title="Leaning Tower of Pisa">tower of Pisa</a>, showing that they both hit the ground at the same time. The reality of that particular experiment is disputed, but he did carry out quantitative experiments by rolling balls on an <a href="/wiki/Inclined_plane" title="Inclined plane">inclined plane</a>. His theory of accelerated motion was derived from the results of such experiments and forms a cornerstone of classical mechanics. In 1673 <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> described in his <i><a href="/wiki/Horologium_Oscillatorium" title="Horologium Oscillatorium">Horologium Oscillatorium</a></i> the first two <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">laws of motion</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The work is also the first modern treatise in which a physical problem (the <a href="/wiki/Acceleration" title="Acceleration">accelerated motion</a> of a falling body) is <a href="/wiki/Mathematical_model" title="Mathematical model">idealized by a set of parameters</a> then analyzed mathematically and constitutes one of the seminal works of <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>.<sup id="cite_ref-:0_22-0" class="reference"><a href="#cite_note-:0-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Portrait_of_Sir_Isaac_Newton,_1689.jpg" class="mw-file-description"><img alt="portrait of Isaac Newton with long hair looking left" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Portrait_of_Sir_Isaac_Newton%2C_1689.jpg/240px-Portrait_of_Sir_Isaac_Newton%2C_1689.jpg" decoding="async" width="240" height="289" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Portrait_of_Sir_Isaac_Newton%2C_1689.jpg/360px-Portrait_of_Sir_Isaac_Newton%2C_1689.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Portrait_of_Sir_Isaac_Newton%2C_1689.jpg/480px-Portrait_of_Sir_Isaac_Newton%2C_1689.jpg 2x" data-file-width="2218" data-file-height="2671" /></a><figcaption>Sir <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> (1643–1727), an influential figure in the history of physics and whose <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">three laws of motion</a> form the basis of classical mechanics</figcaption></figure> <p>Newton founded his principles of natural philosophy on three proposed <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">laws of motion</a>: the <a href="/wiki/Law_of_inertia" class="mw-redirect" title="Law of inertia">law of inertia</a>, his second law of acceleration (mentioned above), and the law of <a href="/wiki/Action_and_reaction" class="mw-redirect" title="Action and reaction">action and reaction</a>; and hence laid the foundations for classical mechanics. Both Newton's second and third laws were given the proper scientific and mathematical treatment in Newton's <i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Philosophiæ Naturalis Principia Mathematica</a></i>. Here they are distinguished from earlier attempts at explaining similar phenomena, which were either incomplete, incorrect, or given little accurate mathematical expression. Newton also enunciated the principles of <a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">conservation of momentum</a> and <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>. In mechanics, Newton was also the first to provide the first correct scientific and mathematical formulation of <a href="/wiki/Gravity" title="Gravity">gravity</a> in <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a>. The combination of Newton's laws of motion and gravitation provides the fullest and most accurate description of classical mechanics. He demonstrated that these laws apply to everyday objects as well as to celestial objects. In particular, he obtained a theoretical explanation of <a href="/wiki/Kepler%27s_laws" class="mw-redirect" title="Kepler's laws">Kepler's laws</a> of motion of the planets. </p><p>Newton had previously invented the <a href="/wiki/Calculus" title="Calculus">calculus</a>; however, the <i>Principia</i> was formulated entirely in terms of long-established geometric methods in emulation of <a href="/wiki/Euclid" title="Euclid">Euclid</a>. Newton, and most of his contemporaries, with the notable exception of <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a>, worked on the assumption that classical mechanics would be able to explain all phenomena, including <a href="/wiki/Light" title="Light">light</a>, in the form of <a href="/wiki/Geometric_optics" class="mw-redirect" title="Geometric optics">geometric optics</a>. Even when discovering the so-called <a href="/wiki/Newton%27s_rings" title="Newton's rings">Newton's rings</a> (a <a href="/wiki/Wave_interference" title="Wave interference">wave interference</a> phenomenon) he maintained his own <a href="/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light">corpuscular theory of light</a>. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Joseph_Louis_Lagrange2.jpg" class="mw-file-description"><img alt="Painting of Joseph-Louis Lagrange" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Joseph_Louis_Lagrange2.jpg/180px-Joseph_Louis_Lagrange2.jpg" decoding="async" width="180" height="249" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Joseph_Louis_Lagrange2.jpg/270px-Joseph_Louis_Lagrange2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Joseph_Louis_Lagrange2.jpg/360px-Joseph_Louis_Lagrange2.jpg 2x" data-file-width="739" data-file-height="1024" /></a><figcaption><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a>'s contribution was realising Newton's ideas in the language of modern mathematics, now called <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>.</figcaption></figure> <p>After Newton, classical mechanics became a principal field of study in mathematics as well as physics. Mathematical formulations progressively allowed finding solutions to a far greater number of problems. The first notable mathematical treatment was in 1788 by <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a>. Lagrangian mechanics was in turn re-formulated in 1833 by <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a>. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:WilliamRowanHamilton.jpeg" class="mw-file-description"><img alt="photograph of William Rowan Hamilton in looking left" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/WilliamRowanHamilton.jpeg/180px-WilliamRowanHamilton.jpeg" decoding="async" width="180" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/81/WilliamRowanHamilton.jpeg 1.5x" data-file-width="268" data-file-height="326" /></a><figcaption><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a> developed an alternative to <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a> now called <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>.</figcaption></figure> <p>Some difficulties were discovered in the late 19th century that could only be resolved by more modern physics. Some of these difficulties related to compatibility with <a href="/wiki/Electromagnetic_theory" class="mw-redirect" title="Electromagnetic theory">electromagnetic theory</a>, and the famous <a href="/wiki/Michelson%E2%80%93Morley_experiment" title="Michelson–Morley experiment">Michelson–Morley experiment</a>. The resolution of these problems led to the <a href="/wiki/Special_theory_of_relativity" class="mw-redirect" title="Special theory of relativity">special theory of relativity</a>, often still considered a part of classical mechanics. </p><p>A second set of difficulties were related to thermodynamics. When combined with <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, classical mechanics leads to the <a href="/wiki/Gibbs_paradox" title="Gibbs paradox">Gibbs paradox</a> of classical <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, in which <a href="/wiki/Entropy" title="Entropy">entropy</a> is not a well-defined quantity. <a href="/wiki/Planck%27s_law" title="Planck's law">Black-body radiation</a> was not explained without the introduction of <a href="/wiki/Quantum" title="Quantum">quanta</a>. As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the <a href="/wiki/Energy_levels" class="mw-redirect" title="Energy levels">energy levels</a> and sizes of <a href="/wiki/Atoms" class="mw-redirect" title="Atoms">atoms</a> and the <a href="/wiki/Photo-electric_effect" class="mw-redirect" title="Photo-electric effect">photo-electric effect</a>. The effort at resolving these problems led to the development of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. </p><p>Since the end of the 20th century, classical mechanics in <a href="/wiki/Physics" title="Physics">physics</a> has no longer been an independent theory. Instead, classical mechanics is now considered an approximate theory to the more general quantum mechanics. Emphasis has shifted to understanding the fundamental forces of nature as in the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> and its more modern extensions into a unified <a href="/wiki/Theory_of_everything" title="Theory of everything">theory of everything</a>. Classical mechanics is a theory useful for the study of the motion of non-quantum mechanical, low-energy particles in weak gravitational fields. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output 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dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/Dynamical_system" title="Dynamical system">Dynamical system</a></li> <li><a href="/wiki/List_of_equations_in_classical_mechanics" title="List of equations in classical mechanics">List of equations in classical mechanics</a></li> <li><a href="/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics#Classical_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">List of publications in classical mechanics</a></li> <li><a href="/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">List of textbooks on classical mechanics and quantum mechanics</a></li> <li><a href="/wiki/Molecular_dynamics" title="Molecular dynamics">Molecular dynamics</a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Classical_mechanics&action=edit&section=19" title="Edit section: Notes"><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 reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">The displacement <span class="texhtml">Δ<b>r</b></span> is the difference of the particle's initial and final positions: <span class="texhtml">Δ<b>r</b> = <b>r</b><sub>final</sub> − <b>r</b><sub>initial</sub></span>.</span> </li> </ol></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=Classical_mechanics&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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="CITEREFBen-Chaim2004" class="citation cs2">Ben-Chaim, Michael (2004), <i>Experimental Philosophy and the Birth of Empirical Science: Boyle, Locke and Newton</i>, Aldershot: Ashgate, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7546-4091-4" title="Special:BookSources/0-7546-4091-4"><bdi>0-7546-4091-4</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/53887772">53887772</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Experimental+Philosophy+and+the+Birth+of+Empirical+Science%3A+Boyle%2C+Locke+and+Newton&rft.place=Aldershot&rft.pub=Ashgate&rft.date=2004&rft_id=info%3Aoclcnum%2F53887772&rft.isbn=0-7546-4091-4&rft.aulast=Ben-Chaim&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAgar2012" class="citation cs2">Agar, Jon (2012), <i>Science in the Twentieth Century and Beyond</i>, Cambridge: Polity Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7456-3469-2" title="Special:BookSources/978-0-7456-3469-2"><bdi>978-0-7456-3469-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+in+the+Twentieth+Century+and+Beyond&rft.place=Cambridge&rft.pub=Polity+Press&rft.date=2012&rft.isbn=978-0-7456-3469-2&rft.aulast=Agar&rft.aufirst=Jon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span>.</span> </li> <li id="cite_note-Wright-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Wright_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Wright_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomas_Wallace_Wright1896" class="citation book cs1">Thomas Wallace Wright (1896). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-LwLAAAAYAAJ&q=mechanics+kinetics"><i>Elements of Mechanics Including Kinematics, Kinetics and Statics: with applications</i></a>. E. and F. N. Spon. p. 85.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Mechanics+Including+Kinematics%2C+Kinetics+and+Statics%3A+with+applications&rft.pages=85&rft.pub=E.+and+F.+N.+Spon&rft.date=1896&rft.au=Thomas+Wallace+Wright&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-LwLAAAAYAAJ%26q%3Dmechanics%2Bkinetics&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Whittaker-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Whittaker_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdmund_Taylor_Whittaker1904" class="citation book cs1"><a href="/wiki/E._T._Whittaker" title="E. T. Whittaker">Edmund Taylor Whittaker</a> (1904). <i><a href="/wiki/A_Treatise_on_the_Analytical_Dynamics_of_Particles_and_Rigid_Bodies" class="mw-redirect" title="A Treatise on the Analytical Dynamics of Particles and Rigid Bodies">A Treatise on the Analytical Dynamics of Particles and Rigid Bodies</a></i>. Cambridge University Press. Chapter 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-35883-3" title="Special:BookSources/0-521-35883-3"><bdi>0-521-35883-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+the+Analytical+Dynamics+of+Particles+and+Rigid+Bodies&rft.pages=Chapter+1&rft.pub=Cambridge+University+Press&rft.date=1904&rft.isbn=0-521-35883-3&rft.au=Edmund+Taylor+Whittaker&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Beggs-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Beggs_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoseph_Stiles_Beggs1983" class="citation book cs1">Joseph Stiles Beggs (1983). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=y6iJ1NIYSmgC"><i>Kinematics</i></a>. Taylor & Francis. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89116-355-7" title="Special:BookSources/0-89116-355-7"><bdi>0-89116-355-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kinematics&rft.pages=1&rft.pub=Taylor+%26+Francis&rft.date=1983&rft.isbn=0-89116-355-7&rft.au=Joseph+Stiles+Beggs&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dy6iJ1NIYSmgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRussell_C._Hibbeler2009" class="citation book cs1">Russell C. Hibbeler (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tOFRjXB-XvMC&pg=PA298">"Kinematics and kinetics of a particle"</a>. <i>Engineering Mechanics: Dynamics</i> (12th ed.). Prentice Hall. p. 298. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-607791-6" title="Special:BookSources/978-0-13-607791-6"><bdi>978-0-13-607791-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Kinematics+and+kinetics+of+a+particle&rft.btitle=Engineering+Mechanics%3A+Dynamics&rft.pages=298&rft.edition=12th&rft.pub=Prentice+Hall&rft.date=2009&rft.isbn=978-0-13-607791-6&rft.au=Russell+C.+Hibbeler&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtOFRjXB-XvMC%26pg%3DPA298&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></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="CITEREFAhmed_A._Shabana2003" class="citation book cs1">Ahmed A. Shabana (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zxuG-l7J5rgC&pg=PA28">"Reference kinematics"</a>. <i>Dynamics of Multibody Systems</i> (2nd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-54411-5" title="Special:BookSources/978-0-521-54411-5"><bdi>978-0-521-54411-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Reference+kinematics&rft.btitle=Dynamics+of+Multibody+Systems&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-54411-5&rft.au=Ahmed+A.+Shabana&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzxuG-l7J5rgC%26pg%3DPA28&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" 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="CITEREFP._P._Teodorescu2007" class="citation book cs1">P. P. Teodorescu (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=k4H2AjWh9qQC&pg=PA287">"Kinematics"</a>. <i>Mechanical Systems, Classical Models: Particle Mechanics</i>. Springer. p. 287. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-5441-9" title="Special:BookSources/978-1-4020-5441-9"><bdi>978-1-4020-5441-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Kinematics&rft.btitle=Mechanical+Systems%2C+Classical+Models%3A+Particle+Mechanics&rft.pages=287&rft.pub=Springer&rft.date=2007&rft.isbn=978-1-4020-5441-9&rft.au=P.+P.+Teodorescu&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dk4H2AjWh9qQC%26pg%3DPA287&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span>.</span> </li> <li id="cite_note-Taylor-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Taylor_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Robert_Taylor2005" class="citation book cs1">John Robert Taylor (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P1kCtNr-pJsC&q=dynamics"><i>Classical Mechanics</i></a>. University Science Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-891389-22-1" title="Special:BookSources/978-1-891389-22-1"><bdi>978-1-891389-22-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.pub=University+Science+Books&rft.date=2005&rft.isbn=978-1-891389-22-1&rft.au=John+Robert+Taylor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DP1kCtNr-pJsC%26q%3Ddynamics&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Greenwood-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Greenwood_10-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDonald_T_Greenwood1997" class="citation book cs1">Donald T Greenwood (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x7rj83I98yMC&q=classical+dynamics"><i>Classical Mechanics</i></a> (Reprint of 1977 ed.). Courier Dover Publications. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-69690-1" title="Special:BookSources/0-486-69690-1"><bdi>0-486-69690-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.pages=1&rft.edition=Reprint+of+1977&rft.pub=Courier+Dover+Publications&rft.date=1997&rft.isbn=0-486-69690-1&rft.au=Donald+T+Greenwood&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dx7rj83I98yMC%26q%3Dclassical%2Bdynamics&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Lanczos-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lanczos_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLanczos1970" class="citation book cs1">Lanczos, Cornelius (1970). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PR4"><i>The variational principles of mechanics</i></a> (4th ed.). New York: Dover Publications Inc. Introduction, pp. xxi–xxix. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-65067-7" title="Special:BookSources/0-486-65067-7"><bdi>0-486-65067-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+variational+principles+of+mechanics&rft.place=New+York&rft.pages=Introduction%2C+pp.+xxi-xxix&rft.edition=4th&rft.pub=Dover+Publications+Inc.&rft.date=1970&rft.isbn=0-486-65067-7&rft.aulast=Lanczos&rft.aufirst=Cornelius&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZWoYYr8wk2IC%26pg%3DPR4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnudsenHjorth2012" class="citation book cs1">Knudsen, Jens M.; Hjorth, Poul (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rkP1CAAAQBAJ"><i>Elements of Newtonian Mechanics</i></a> (illustrated ed.). Springer Science & Business Media. p. 30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-97599-8" title="Special:BookSources/978-3-642-97599-8"><bdi>978-3-642-97599-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Newtonian+Mechanics&rft.pages=30&rft.edition=illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-3-642-97599-8&rft.aulast=Knudsen&rft.aufirst=Jens+M.&rft.au=Hjorth%2C+Poul&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrkP1CAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rkP1CAAAQBAJ&pg=PA30">Extract of page 30</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://ocw.mit.edu/courses/physics/8-01-physics-i-fall-2003/lecture-notes/binder1.pdf">MIT physics 8.01 lecture notes (page 12)</a>. <a rel="nofollow" class="external text" href="http://webarchive.loc.gov/all/20130709154423/http%3A//ocw.mit.edu/courses/physics/8%2D01%2Dphysics%2Di%2Dfall%2D2003/lecture%2Dnotes/binder1.pdf">Archived</a> 2013-07-09 at the <a href="/wiki/Library_of_Congress" title="Library of Congress">Library of Congress</a> Web Archives (PDF)</span> </li> <li id="cite_note-Goldstein-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Goldstein_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Goldstein_14-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Goldstein, Herbert (1950). Classical Mechanics (1st ed.). Addison-Wesley.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThorntonMarion2004" class="citation book cs1">Thornton, Stephen T.; Marion, Jerry B. (2004). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/classicaldynamic00thor"><i>Classical dynamics of particles and systems</i></a></span> (5. ed.). Belmont, CA: Brooks/Cole. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/classicaldynamic00thor/page/n67">50</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-40896-1" title="Special:BookSources/978-0-534-40896-1"><bdi>978-0-534-40896-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+dynamics+of+particles+and+systems&rft.place=Belmont%2C+CA&rft.pages=50&rft.edition=5.&rft.pub=Brooks%2FCole&rft.date=2004&rft.isbn=978-0-534-40896-1&rft.aulast=Thornton&rft.aufirst=Stephen+T.&rft.au=Marion%2C+Jerry+B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fclassicaldynamic00thor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFraser1983" class="citation journal cs1">Fraser, Craig (1983). "J. L. Lagrange's Early Contributions to the Principles and Methods of Mechanics". <i>Archive for History of Exact Sciences</i>. <b>28</b> (3): 197–241. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00328268">10.1007/BF00328268</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41133689">41133689</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=J.+L.+Lagrange%27s+Early+Contributions+to+the+Principles+and+Methods+of+Mechanics&rft.volume=28&rft.issue=3&rft.pages=197-241&rft.date=1983&rft_id=info%3Adoi%2F10.1007%2FBF00328268&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F41133689%23id-name%3DJSTOR&rft.aulast=Fraser&rft.aufirst=Craig&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHandFinch1998" class="citation book cs1">Hand, L. N.; Finch, J. D. (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA141"><i>Analytical Mechanics</i></a> (2nd ed.). Cambridge University Press. pp. 18–20, 23, 46, 51. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521575720" title="Special:BookSources/9780521575720"><bdi>9780521575720</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytical+Mechanics&rft.pages=18-20%2C+23%2C+46%2C+51&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1998&rft.isbn=9780521575720&rft.aulast=Hand&rft.aufirst=L.+N.&rft.au=Finch%2C+J.+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1J2hzvX2Xh8C%26pg%3DPA141&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamilton,_William_Rowan1833" class="citation book cs1">Hamilton, William Rowan (1833). <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/68159539"><i>On a general method of expressing the paths of light, & of the planets, by the coefficients of a characteristic function</i></a>. Printed by P.D. Hardy. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/68159539">68159539</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+a+general+method+of+expressing+the+paths+of+light%2C+%26+of+the+planets%2C+by+the+coefficients+of+a+characteristic+function.&rft.pub=Printed+by+P.D.+Hardy&rft.date=1833&rft_id=info%3Aoclcnum%2F68159539&rft.au=Hamilton%2C+William+Rowan&rft_id=http%3A%2F%2Fworldcat.org%2Foclc%2F68159539&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDoranLasenby2003" class="citation book cs1">Doran, Chris; Lasenby, Anthony N. (2003). <i>Geometric algebra for physicists</i>. Cambridge New York: Cambridge university press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-48022-2" title="Special:BookSources/978-0-521-48022-2"><bdi>978-0-521-48022-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+algebra+for+physicists&rft.place=Cambridge+New+York&rft.pub=Cambridge+university+press&rft.date=2003&rft.isbn=978-0-521-48022-2&rft.aulast=Doran&rft.aufirst=Chris&rft.au=Lasenby%2C+Anthony+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRob_Iliffe_&_George_E._Smith2016" class="citation book cs1">Rob Iliffe & George E. Smith (2016). <i>The Cambridge Companion to Newton</i>. Cambridge University Press. p. 75. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781107015463" title="Special:BookSources/9781107015463"><bdi>9781107015463</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Cambridge+Companion+to+Newton&rft.pages=75&rft.pub=Cambridge+University+Press&rft.date=2016&rft.isbn=9781107015463&rft.au=Rob+Iliffe+%26+George+E.+Smith&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-:0-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoder1988" class="citation book cs1"><a href="/wiki/Joella_Yoder" title="Joella Yoder">Yoder, Joella G.</a> (1988). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/books/unrolling-time/1427509C7A14C464B08209322E42ABB6"><i>Unrolling Time: Christiaan Huygens and the Mathematization of Nature</i></a>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-34140-0" title="Special:BookSources/978-0-521-34140-0"><bdi>978-0-521-34140-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unrolling+Time%3A+Christiaan+Huygens+and+the+Mathematization+of+Nature&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1988&rft.isbn=978-0-521-34140-0&rft.aulast=Yoder&rft.aufirst=Joella+G.&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fbooks%2Funrolling-time%2F1427509C7A14C464B08209322E42ABB6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" 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=Classical_mechanics&action=edit&section=21" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlonso,_M.Finn,_J.1992" class="citation book cs1">Alonso, M.; Finn, J. (1992). <i>Fundamental University Physics</i>. Addison-Wesley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+University+Physics&rft.pub=Addison-Wesley&rft.date=1992&rft.au=Alonso%2C+M.&rft.au=Finn%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman,_Richard1999" class="citation book cs1"><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman, Richard</a> (1999). <a href="/wiki/The_Feynman_Lectures_on_Physics" title="The Feynman Lectures on Physics"><i>The Feynman Lectures on Physics</i></a>. Perseus Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7382-0092-7" title="Special:BookSources/978-0-7382-0092-7"><bdi>978-0-7382-0092-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+Lectures+on+Physics&rft.pub=Perseus+Publishing&rft.date=1999&rft.isbn=978-0-7382-0092-7&rft.au=Feynman%2C+Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman,_RichardPhillips,_Richard1998" class="citation book cs1">Feynman, Richard; Phillips, Richard (1998). <i>Six Easy Pieces</i>. Perseus Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-32841-7" title="Special:BookSources/978-0-201-32841-7"><bdi>978-0-201-32841-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Six+Easy+Pieces&rft.pub=Perseus+Publishing&rft.date=1998&rft.isbn=978-0-201-32841-7&rft.au=Feynman%2C+Richard&rft.au=Phillips%2C+Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldstein,_HerbertCharles_P._PooleJohn_L._Safko2002" class="citation book cs1"><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Goldstein, Herbert</a>; Charles P. Poole; John L. Safko (2002). <a href="/wiki/Classical_Mechanics_(Goldstein_book)" class="mw-redirect" title="Classical Mechanics (Goldstein book)"><i>Classical Mechanics</i></a> (3rd ed.). Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-65702-9" title="Special:BookSources/978-0-201-65702-9"><bdi>978-0-201-65702-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.edition=3rd&rft.pub=Addison+Wesley&rft.date=2002&rft.isbn=978-0-201-65702-9&rft.au=Goldstein%2C+Herbert&rft.au=Charles+P.+Poole&rft.au=John+L.+Safko&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKibbleBerkshire2004" class="citation book cs1"><a href="/wiki/Tom_Kibble" title="Tom Kibble">Kibble, Tom W.B.</a>; <a href="/wiki/Frank_H._Berkshire" title="Frank H. Berkshire">Berkshire, Frank H.</a> (2004). <a href="/wiki/Classical_Mechanics_(Kibble_and_Berkshire)" title="Classical Mechanics (Kibble and Berkshire)"><i>Classical Mechanics (5th ed.)</i></a>. <a href="/wiki/Imperial_College_Press" title="Imperial College Press">Imperial College Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-86094-424-6" title="Special:BookSources/978-1-86094-424-6"><bdi>978-1-86094-424-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics+%285th+ed.%29&rft.pub=Imperial+College+Press&rft.date=2004&rft.isbn=978-1-86094-424-6&rft.aulast=Kibble&rft.aufirst=Tom+W.B.&rft.au=Berkshire%2C+Frank+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleppner,_D.Kolenkow,_R.J.1973" class="citation book cs1">Kleppner, D.; Kolenkow, R.J. (1973). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontome00dani"><i>An Introduction to Mechanics</i></a>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-035048-9" title="Special:BookSources/978-0-07-035048-9"><bdi>978-0-07-035048-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Mechanics&rft.pub=McGraw-Hill&rft.date=1973&rft.isbn=978-0-07-035048-9&rft.au=Kleppner%2C+D.&rft.au=Kolenkow%2C+R.J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontome00dani&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandau,_L.D.Lifshitz,_E.M.1972" class="citation book cs1">Landau, L.D.; Lifshitz, E.M. (1972). <a href="/wiki/Course_of_Theoretical_Physics" title="Course of Theoretical Physics"><i>Course of Theoretical Physics, Vol. 1 – Mechanics</i></a>. Franklin Book Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-016739-8" title="Special:BookSources/978-0-08-016739-8"><bdi>978-0-08-016739-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Course+of+Theoretical+Physics%2C+Vol.+1+%E2%80%93+Mechanics&rft.pub=Franklin+Book+Company&rft.date=1972&rft.isbn=978-0-08-016739-8&rft.au=Landau%2C+L.D.&rft.au=Lifshitz%2C+E.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorin2008" class="citation book cs1">Morin, David (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontocl00mori"><i>Introduction to Classical Mechanics: With Problems and Solutions</i></a> (1st ed.). Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-87622-3" title="Special:BookSources/978-0-521-87622-3"><bdi>978-0-521-87622-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Classical+Mechanics%3A+With+Problems+and+Solutions&rft.place=Cambridge&rft.edition=1st&rft.pub=Cambridge+University+Press&rft.date=2008&rft.isbn=978-0-521-87622-3&rft.aulast=Morin&rft.aufirst=David&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontocl00mori&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGerald_Jay_SussmanJack_Wisdom2001" class="citation book cs1"><a href="/wiki/Gerald_Jay_Sussman" title="Gerald Jay Sussman">Gerald Jay Sussman</a>; <a href="/wiki/Jack_Wisdom" title="Jack Wisdom">Jack Wisdom</a> (2001). <a href="/wiki/Structure_and_Interpretation_of_Classical_Mechanics" title="Structure and Interpretation of Classical Mechanics"><i>Structure and Interpretation of Classical Mechanics</i></a>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-19455-6" title="Special:BookSources/978-0-262-19455-6"><bdi>978-0-262-19455-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Structure+and+Interpretation+of+Classical+Mechanics&rft.pub=MIT+Press&rft.date=2001&rft.isbn=978-0-262-19455-6&rft.au=Gerald+Jay+Sussman&rft.au=Jack+Wisdom&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'Donnell,_Peter_J.2015" class="citation book cs1">O'Donnell, Peter J. (2015). <i>Essential Dynamics and Relativity</i>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4665-8839-4" title="Special:BookSources/978-1-4665-8839-4"><bdi>978-1-4665-8839-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essential+Dynamics+and+Relativity&rft.pub=CRC+Press&rft.date=2015&rft.isbn=978-1-4665-8839-4&rft.au=O%27Donnell%2C+Peter+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThornton,_Stephen_T.Marion,_Jerry_B.2003" class="citation book cs1">Thornton, Stephen T.; Marion, Jerry B. (2003). <i>Classical Dynamics of Particles and Systems (5th ed.)</i>. Brooks Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-40896-1" title="Special:BookSources/978-0-534-40896-1"><bdi>978-0-534-40896-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Dynamics+of+Particles+and+Systems+%285th+ed.%29&rft.pub=Brooks+Cole&rft.date=2003&rft.isbn=978-0-534-40896-1&rft.au=Thornton%2C+Stephen+T.&rft.au=Marion%2C+Jerry+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AClassical+mechanics" class="Z3988"></span></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=Classical_mechanics&action=edit&section=22" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Classical_mechanics" class="extiw" title="commons:Category:Classical mechanics">Classical mechanics</a></span>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></span></span></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Classical_mechanics" class="extiw" title="q:Special:Search/Classical mechanics">Classical mechanics</a></b></i>.</div></div> </div> <ul><li>Crowell, Benjamin. <a rel="nofollow" class="external text" href="http://www.lightandmatter.com/lm">Light and Matter</a> (an introductory text, uses algebra with optional sections involving calculus)</li> <li>Fitzpatrick, Richard. <a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/301/301.html">Classical Mechanics</a> (uses calculus)</li> <li>Hoiland, Paul (2004). <a rel="nofollow" class="external text" href="http://doc.cern.ch//archive/electronic/other/ext/ext-2004-126.pdf">Preferred Frames of Reference & Relativity</a></li> <li>Horbatsch, Marko, "<i><a rel="nofollow" class="external text" href="http://www.yorku.ca/marko/PHYS2010/index.htm">Classical Mechanics Course Notes</a></i>".</li> <li>Rosu, Haret C., "<i><a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/9909035">Classical Mechanics</a></i>". Physics Education. 1999. [arxiv.org : physics/9909035]</li> <li>Shapiro, Joel A. (2003). <a rel="nofollow" class="external text" href="http://www.physics.rutgers.edu/ugrad/494/bookr03D.pdf">Classical Mechanics</a></li> <li>Sussman, Gerald Jay & Wisdom, Jack & Mayer, Meinhard E. (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120920024409/http://mitpress.mit.edu/SICM/">Structure and Interpretation of Classical Mechanics</a></li> <li>Tong, David. <a rel="nofollow" class="external text" href="http://www.damtp.cam.ac.uk/user/tong/dynamics.html">Classical Dynamics</a> (Cambridge lecture notes on Lagrangian and Hamiltonian formalism)</li> <li><a rel="nofollow" class="external text" href="http://kmoddl.library.cornell.edu/index.php">Kinematic Models for Design Digital Library (KMODDL)</a><br /> Movies and photos of hundreds of working mechanical-systems models at <a href="/wiki/Cornell_University" title="Cornell University">Cornell University</a>. Also includes an <a rel="nofollow" class="external text" href="http://kmoddl.library.cornell.edu/e-books.php">e-book library</a> of classic texts on mechanical design and engineering.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20190327102351/https://ocw.mit.edu/courses/physics/8-01sc-physics-i-classical-mechanics-fall-2010/">MIT OpenCourseWare 8.01: Classical Mechanics</a> Free videos of actual course lectures with links to lecture notes, assignments and exams.</li> <li>Alejandro A. Torassa, <a rel="nofollow" class="external text" href="http://torassa.tripod.com/paper.htm">On Classical Mechanics</a></li></ul> <div style="clear:both;" class=""></div> <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><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"></div><div role="navigation" class="navbox" aria-labelledby="Sir_Isaac_Newton" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Isaac_Newton" title="Template:Isaac Newton"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Isaac_Newton" title="Template talk:Isaac Newton"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Isaac_Newton" title="Special:EditPage/Template:Isaac Newton"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Sir_Isaac_Newton" style="font-size:114%;margin:0 4em"><a href="/wiki/Isaac_Newton" title="Isaac Newton">Sir Isaac Newton</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Publications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Fluxions</a></i> (1671)</li> <li><i><a href="/wiki/De_motu_corporum_in_gyrum" title="De motu corporum in gyrum">De Motu</a></i> (1684)</li> <li><i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i> (1687)</li> <li><i><a href="/wiki/Opticks" title="Opticks">Opticks</a></i> (1704)</li> <li><i><a href="/wiki/The_Queries" class="mw-redirect" title="The Queries">Queries</a></i> (1704)</li> <li><i><a href="/wiki/Arithmetica_Universalis" title="Arithmetica Universalis">Arithmetica</a></i> (1707)</li> <li><i><a href="/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas" title="De analysi per aequationes numero terminorum infinitas">De Analysi</a></i> (1711)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Other writings</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Quaestiones_quaedam_philosophicae" title="Quaestiones quaedam philosophicae">Quaestiones</a></i> (1661–1665)</li> <li>"<a href="/wiki/Standing_on_the_shoulders_of_giants" title="Standing on the shoulders of giants">standing on the shoulders of giants</a>" (1675)</li> <li><i><a href="/wiki/Notes_on_the_Jewish_Temple" title="Notes on the Jewish Temple">Notes on the Jewish Temple</a></i> (c. 1680)</li> <li>"<a href="/wiki/General_Scholium" title="General Scholium">General Scholium</a>" (1713; <i>"<a href="/wiki/Hypotheses_non_fingo" title="Hypotheses non fingo">hypotheses non fingo</a>"</i> )</li> <li><i><a href="/wiki/The_Chronology_of_Ancient_Kingdoms_Amended" title="The Chronology of Ancient Kingdoms Amended">Ancient Kingdoms Amended</a></i> (1728)</li> <li><i><a href="/wiki/An_Historical_Account_of_Two_Notable_Corruptions_of_Scripture" title="An Historical Account of Two Notable Corruptions of Scripture">Corruptions of Scripture</a></i> (1754)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Contributions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a> <ul><li><a href="/wiki/Fluxion" title="Fluxion">fluxion</a></li></ul></li> <li><a href="/wiki/Impact_depth" title="Impact depth">Impact depth</a></li> <li><a href="/wiki/Inertia" title="Inertia">Inertia</a></li> <li><a href="/wiki/Newton_disc" title="Newton disc">Newton disc</a></li> <li><a href="/wiki/Newton_polygon" title="Newton polygon">Newton polygon</a> <ul><li><a href="/wiki/Newton%E2%80%93Okounkov_body" title="Newton–Okounkov body">Newton–Okounkov body</a></li></ul></li> <li><a href="/wiki/Newton%27s_reflector" title="Newton's reflector">Newton's reflector</a></li> <li><a href="/wiki/Newtonian_telescope" title="Newtonian telescope">Newtonian telescope</a></li> <li><a href="/wiki/Newton_scale" title="Newton scale">Newton scale</a></li> <li><a href="/wiki/Newton%27s_metal" title="Newton's metal">Newton's metal</a></li> <li><a href="/wiki/Spectrum" title="Spectrum">Spectrum</a></li> <li><a href="/wiki/Structural_coloration" title="Structural coloration">Structural coloration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Newtonianism" title="Newtonianism">Newtonianism</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bucket_argument" title="Bucket argument">Bucket argument</a></li> <li><a href="/wiki/Newton%27s_inequalities" title="Newton's inequalities">Newton's inequalities</a></li> <li><a href="/wiki/Newton%27s_law_of_cooling" title="Newton's law of cooling">Newton's law of cooling</a></li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a> <ul><li><a href="/wiki/Post-Newtonian_expansion" title="Post-Newtonian expansion">post-Newtonian expansion</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">parameterized</a></li> <li><a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a></li></ul></li> <li><a href="/wiki/Newton%E2%80%93Cartan_theory" title="Newton–Cartan theory">Newton–Cartan theory</a></li> <li><a href="/wiki/Schr%C3%B6dinger%E2%80%93Newton_equation" title="Schrödinger–Newton equation">Schrödinger–Newton equation</a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> <ul><li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws</a></li></ul></li> <li><a href="/wiki/Newtonian_dynamics" title="Newtonian dynamics">Newtonian dynamics</a></li> <li><a href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method in optimization</a> <ul><li><a href="/wiki/Problem_of_Apollonius" title="Problem of Apollonius">Apollonius's problem</a></li> <li><a href="/wiki/Truncated_Newton_method" title="Truncated Newton method">truncated Newton method</a></li></ul></li> <li><a href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton algorithm</a></li> <li><a href="/wiki/Newton%27s_rings" title="Newton's rings">Newton's rings</a></li> <li><a href="/wiki/Newton%27s_theorem_about_ovals" title="Newton's theorem about ovals">Newton's theorem about ovals</a></li> <li><a href="/wiki/Newton%E2%80%93Pepys_problem" title="Newton–Pepys problem">Newton–Pepys problem</a></li> <li><a href="/wiki/Newtonian_potential" title="Newtonian potential">Newtonian potential</a></li> <li><a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</a></li> <li><a class="mw-selflink selflink">Classical mechanics</a></li> <li><a href="/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light">Corpuscular theory of light</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton calculus controversy</a></li> <li><a href="/wiki/Newton%27s_notation" class="mw-redirect" title="Newton's notation">Newton's notation</a></li> <li><a href="/wiki/Rotating_spheres" title="Rotating spheres">Rotating spheres</a></li> <li><a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a></li> <li><a href="/wiki/Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes formulas</a></li> <li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> <ul><li><a href="/wiki/Generalized_Gauss%E2%80%93Newton_method" title="Generalized Gauss–Newton method">generalized Gauss–Newton method</a></li></ul></li> <li><a href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</a></li> <li><a href="/wiki/Newton%27s_identities" title="Newton's identities">Newton's identities</a></li> <li><a href="/wiki/Newton_polynomial" title="Newton polynomial">Newton polynomial</a></li> <li><a href="/wiki/Newton%27s_theorem_of_revolving_orbits" title="Newton's theorem of revolving orbits">Newton's theorem of revolving orbits</a></li> <li><a href="/wiki/Newton%E2%80%93Euler_equations" title="Newton–Euler equations">Newton–Euler equations</a></li> <li><a href="/wiki/Power_number" title="Power number">Newton number</a> <ul><li><a href="/wiki/Kissing_number" title="Kissing number">kissing number problem</a></li></ul></li> <li><a href="/wiki/Difference_quotient" title="Difference quotient">Newton's quotient</a></li> <li><a href="/wiki/Parallelogram_of_force" title="Parallelogram of force">Parallelogram of force</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Newton–Puiseux theorem</a></li> <li><a href="/wiki/Absolute_space_and_time#Newton" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Luminiferous_aether" title="Luminiferous aether">Luminiferous aether</a></li> <li><a href="/wiki/Finite_difference" title="Finite difference">Newtonian series</a> <ul><li><a href="/wiki/Table_of_Newtonian_series" title="Table of Newtonian series">table</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Personal life</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Woolsthorpe_Manor" title="Woolsthorpe Manor">Woolsthorpe Manor</a> (birthplace)</li> <li><a href="/wiki/Cranbury_Park" title="Cranbury Park">Cranbury Park</a> (home)</li> <li><a href="/wiki/Early_life_of_Isaac_Newton" title="Early life of Isaac Newton">Early life</a></li> <li><a href="/wiki/Later_life_of_Isaac_Newton" title="Later life of Isaac Newton">Later life</a></li> <li><a href="/wiki/Isaac_Newton%27s_apple_tree" title="Isaac Newton's apple tree">Apple tree</a></li> <li><a href="/wiki/Religious_views_of_Isaac_Newton" title="Religious views of Isaac Newton">Religious views</a></li> <li><a href="/wiki/Isaac_Newton%27s_occult_studies" title="Isaac Newton's occult studies">Occult studies</a></li> <li><a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a></li> <li><a href="/wiki/Copernican_Revolution" title="Copernican Revolution">Copernican Revolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Relations</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Catherine_Barton" title="Catherine Barton">Catherine Barton</a> (niece)</li> <li><a href="/wiki/John_Conduitt" title="John Conduitt">John Conduitt</a> (nephew-in-law)</li> <li><a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Isaac Barrow</a> (professor)</li> <li><a href="/wiki/William_Clarke_(apothecary)" title="William Clarke (apothecary)">William Clarke</a> (mentor)</li> <li><a href="/wiki/Benjamin_Pulleyn" title="Benjamin Pulleyn">Benjamin Pulleyn</a> (tutor)</li> <li><a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a> (student)</li> <li><a href="/wiki/William_Whiston" title="William Whiston">William Whiston</a> (student)</li> <li><a href="/wiki/John_Keill" title="John Keill">John Keill</a> (disciple)</li> <li><a href="/wiki/William_Stukeley" title="William Stukeley">William Stukeley</a> (friend)</li> <li><a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a> (friend)</li> <li><a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> (friend)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Isaac_Newton_in_popular_culture" title="Isaac Newton in popular culture">Depictions</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(Blake)" title="Newton (Blake)"><i>Newton</i> by Blake</a> (monotype)</li> <li><a href="/wiki/Newton_(Paolozzi)" title="Newton (Paolozzi)"><i>Newton</i> by Paolozzi</a> (sculpture)</li> <li><i><a href="/wiki/Isaac_Newton_Gargoyle" title="Isaac Newton Gargoyle">Isaac Newton Gargoyle</a></i></li> <li><i><a href="/wiki/Astronomers_Monument" title="Astronomers Monument">Astronomers Monument</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/List_of_things_named_after_Isaac_Newton" title="List of things named after Isaac Newton">Namesake</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(unit)" title="Newton (unit)">Newton (unit)</a></li> <li><a href="/wiki/Newton%27s_cradle" title="Newton's cradle">Newton's cradle</a></li> <li><a href="/wiki/Isaac_Newton_Institute" title="Isaac Newton Institute">Isaac Newton Institute</a></li> <li><a href="/wiki/Institute_of_Physics_Isaac_Newton_Medal" class="mw-redirect" title="Institute of Physics Isaac Newton Medal">Isaac Newton Medal</a></li> <li><a href="/wiki/Isaac_Newton_Telescope" title="Isaac Newton Telescope">Isaac Newton Telescope</a></li> <li><a href="/wiki/Isaac_Newton_Group_of_Telescopes" title="Isaac Newton Group of Telescopes">Isaac Newton Group of Telescopes</a></li> <li><a href="/wiki/XMM-Newton" title="XMM-Newton">XMM-Newton</a></li> <li><a href="/wiki/Sir_Isaac_Newton_Sixth_Form" title="Sir Isaac Newton Sixth Form">Sir Isaac Newton Sixth Form</a></li> <li><a href="/wiki/Statal_Institute_of_Higher_Education_Isaac_Newton" title="Statal Institute of Higher Education Isaac Newton">Statal Institute of Higher Education Isaac Newton</a></li> <li><a href="/wiki/Newton_International_Fellowship" title="Newton International Fellowship">Newton International Fellowship</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Categories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><div class="div-col"> <div class="CategoryTreeTag" data-ct-options="{"mode":20,"hideprefix":20,"showcount":false,"namespaces":false,"notranslations":false}"><div class="CategoryTreeSection"><div class="CategoryTreeItem"><span class="CategoryTreeBullet"><a class="CategoryTreeToggle" data-ct-title="Isaac_Newton" aria-expanded="false"></a> </span> <bdi dir="ltr"><a href="/wiki/Category:Isaac_Newton" title="Category:Isaac Newton">Isaac Newton</a></bdi></div><div class="CategoryTreeChildren" style="display:none"></div></div></div> </div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_branches_of_physics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Branches_of_physics" title="Template:Branches of physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Branches_of_physics" title="Template talk:Branches of physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Branches_of_physics" title="Special:EditPage/Template:Branches of physics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_branches_of_physics" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Branches_of_physics" title="Branches of physics">branches of physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Basic_research" title="Basic research">Pure</a></li> <li><a href="/wiki/Applied_physics" title="Applied physics">Applied</a> <ul><li><a href="/wiki/Engineering_physics" title="Engineering physics">Engineering</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Approaches</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Experimental_physics" title="Experimental physics">Experimental</a></li> <li><a href="/wiki/Theoretical_physics" title="Theoretical physics">Theoretical</a> <ul><li><a href="/wiki/Computational_physics" title="Computational physics">Computational</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Classical_physics" title="Classical physics">Classical</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Classical mechanics</a> <ul><li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newtonian</a></li> <li><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li></ul></li> <li><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></li> <li><a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">Classical electromagnetism</a></li> <li><a href="/wiki/Classical_optics" class="mw-redirect" title="Classical optics">Classical optics</a> <ul><li><a href="/wiki/Geometrical_optics" title="Geometrical optics">Ray</a></li> <li><a href="/wiki/Physical_optics" title="Physical optics">Wave</a></li></ul></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a> <ul><li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Modern_physics" title="Modern physics">Modern</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Relativistic_mechanics" title="Relativistic mechanics">Relativistic mechanics</a> <ul><li><a href="/wiki/Special_relativity" title="Special relativity">Special</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General</a></li></ul></li> <li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li> <li><a href="/wiki/Particle_physics" title="Particle physics">Particle physics</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Atomic,_molecular,_and_optical_physics" title="Atomic, molecular, and optical physics">Atomic, molecular, and optical physics</a> <ul><li><a href="/wiki/Atomic_physics" title="Atomic physics">Atomic</a></li> <li><a href="/wiki/Molecular_physics" title="Molecular physics">Molecular</a></li> <li><a href="/wiki/Optics#Modern_optics" title="Optics">Modern optics</a></li></ul></li> <li><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a> <ul><li><a href="/wiki/Solid-state_physics" title="Solid-state physics">Solid-state physics</a></li> <li><a href="/wiki/Crystallography" title="Crystallography">Crystallography</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Applied_and_interdisciplinary_physics" title="Category:Applied and interdisciplinary physics">Interdisciplinary</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astrophysics" title="Astrophysics">Astrophysics</a></li> <li><a href="/wiki/Atmospheric_physics" title="Atmospheric physics">Atmospheric physics</a></li> <li><a href="/wiki/Biophysics" title="Biophysics">Biophysics</a></li> <li><a href="/wiki/Chemical_physics" title="Chemical physics">Chemical physics</a></li> <li><a href="/wiki/Geophysics" title="Geophysics">Geophysics</a></li> <li><a href="/wiki/Materials_science" title="Materials science">Materials science</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Medical_physics" title="Medical physics">Medical physics</a></li> <li><a href="/wiki/Physical_oceanography" title="Physical oceanography">Ocean physics</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_physics" title="History of physics">History of physics</a></li> <li><a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a></li> <li><a href="/wiki/Philosophy_of_physics" title="Philosophy of physics">Philosophy of physics</a></li> <li><a href="/wiki/Physics_education" title="Physics education">Physics education</a></li> <li><a href="/wiki/Timeline_of_fundamental_physics_discoveries" title="Timeline of fundamental physics discoveries">Timeline of physics discoveries</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" 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