Lagrange-mekanikk – Wikipedia

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data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Innhold</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">flytt til sidefeltet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skjul</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Til toppen)</div> </a> </li> <li id="toc-Generelle_koordinater" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generelle_koordinater"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Generelle koordinater</span> </div> </a> <button aria-controls="toc-Generelle_koordinater-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>Vis/skjul underseksjonen Generelle koordinater</span> </button> <ul id="toc-Generelle_koordinater-sublist" class="vector-toc-list"> <li id="toc-Kinematiske_relasjoner" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kinematiske_relasjoner"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Kinematiske relasjoner</span> </div> </a> <ul id="toc-Kinematiske_relasjoner-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-D'Alemberts_prinsipp" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#D'Alemberts_prinsipp"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>D'Alemberts prinsipp</span> </div> </a> <ul id="toc-D'Alemberts_prinsipp-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lagrange-funksjonen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lagrange-funksjonen"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Lagrange-funksjonen</span> </div> </a> <ul id="toc-Lagrange-funksjonen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bevegelseskonstanter" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bevegelseskonstanter"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Bevegelseskonstanter</span> </div> </a> <button aria-controls="toc-Bevegelseskonstanter-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>Vis/skjul underseksjonen Bevegelseskonstanter</span> </button> <ul id="toc-Bevegelseskonstanter-sublist" class="vector-toc-list"> <li id="toc-Hamilton-funksjonen" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hamilton-funksjonen"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Hamilton-funksjonen</span> </div> </a> <ul id="toc-Hamilton-funksjonen-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sentralsymmetrisk_potensial" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sentralsymmetrisk_potensial"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Sentralsymmetrisk potensial</span> </div> </a> <ul id="toc-Sentralsymmetrisk_potensial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geodetisk_bevegelse" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geodetisk_bevegelse"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Geodetisk bevegelse</span> </div> </a> <button aria-controls="toc-Geodetisk_bevegelse-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>Vis/skjul underseksjonen Geodetisk bevegelse</span> </button> <ul id="toc-Geodetisk_bevegelse-sublist" class="vector-toc-list"> <li id="toc-Eksempel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eksempel"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Eksempel</span> </div> </a> <ul id="toc-Eksempel-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Se_også" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Se_også"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Se også</span> </div> </a> <ul id="toc-Se_også-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referanser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referanser"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Referanser</span> </div> </a> <ul id="toc-Referanser-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Litteratur" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Litteratur"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Litteratur</span> </div> </a> <ul id="toc-Litteratur-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="Innhold" 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="Vis/skjul innholdsfortegnelsen" > <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">Vis/skjul innholdsfortegnelsen</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">Lagrange-mekanikk</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="Gå til en artikkel på et annet språk. Tilgjengelig på 40 språk" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-40" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">40 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Lagrangemekanikk" title="Lagrangemekanikk – norsk nynorsk" lang="nn" hreflang="nn" data-title="Lagrangemekanikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norsk nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Lagranges_ekvationer" title="Lagranges ekvationer – svensk" lang="sv" hreflang="sv" data-title="Lagranges ekvationer" data-language-autonym="Svenska" data-language-local-name="svensk" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Lagrange-meganika" title="Lagrange-meganika – afrikaans" lang="af" hreflang="af" data-title="Lagrange-meganika" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</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_%D9%84%D8%A7%D8%BA%D8%B1%D8%A7%D9%86%D8%AC" title="ميكانيكا لاغرانج – arabisk" lang="ar" hreflang="ar" data-title="ميكانيكا لاغرانج" data-language-autonym="العربية" data-language-local-name="arabisk" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%B0%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Лагранжава механіка – belarusisk" lang="be" hreflang="be" data-title="Лагранжава механіка" data-language-autonym="Беларуская" data-language-local-name="belarusisk" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0_%D0%BD%D0%B0_%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6" title="Механика на Лагранж – bulgarsk" lang="bg" hreflang="bg" data-title="Механика на Лагранж" data-language-autonym="Български" data-language-local-name="bulgarsk" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Formulaci%C3%B3_lagrangiana" title="Formulació lagrangiana – katalansk" lang="ca" hreflang="ca" data-title="Formulació lagrangiana" data-language-autonym="Català" data-language-local-name="katalansk" 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%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%BB%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Лагранжла механика – tsjuvasjisk" lang="cv" hreflang="cv" data-title="Лагранжла механика" data-language-autonym="Чӑвашла" data-language-local-name="tsjuvasjisk" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lagrange-Formalismus" title="Lagrange-Formalismus – tysk" lang="de" hreflang="de" data-title="Lagrange-Formalismus" data-language-autonym="Deutsch" data-language-local-name="tysk" 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/Lagrange%27i_mehaanika" title="Lagrange'i mehaanika – estisk" lang="et" hreflang="et" data-title="Lagrange'i mehaanika" data-language-autonym="Eesti" data-language-local-name="estisk" 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%9B%CE%B1%CE%B3%CE%BA%CF%81%CE%B1%CE%BD%CE%B6%CE%B9%CE%B1%CE%BD%CE%AE_%CE%BC%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE" title="Λαγκρανζιανή μηχανική – gresk" lang="el" hreflang="el" data-title="Λαγκρανζιανή μηχανική" data-language-autonym="Ελληνικά" data-language-local-name="gresk" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Lagrangian_mechanics" title="Lagrangian mechanics – engelsk" lang="en" hreflang="en" data-title="Lagrangian mechanics" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Mec%C3%A1nica_lagrangiana" title="Mecánica lagrangiana – spansk" lang="es" hreflang="es" data-title="Mecánica lagrangiana" data-language-autonym="Español" data-language-local-name="spansk" 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/Lagran%C4%9Da_mekaniko" title="Lagranĝa mekaniko – esperanto" lang="eo" hreflang="eo" data-title="Lagranĝa mekaniko" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Lagrangeren_mekanika" title="Lagrangeren mekanika – baskisk" lang="eu" hreflang="eu" data-title="Lagrangeren mekanika" data-language-autonym="Euskara" data-language-local-name="baskisk" 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_%D9%84%D8%A7%DA%AF%D8%B1%D8%A7%D9%86%DA%98%DB%8C" title="مکانیک لاگرانژی – persisk" lang="fa" hreflang="fa" data-title="مکانیک لاگرانژی" data-language-autonym="فارسی" data-language-local-name="persisk" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quations_de_Lagrange" title="Équations de Lagrange – fransk" lang="fr" hreflang="fr" data-title="Équations de Lagrange" data-language-autonym="Français" data-language-local-name="fransk" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Mec%C3%A1nica_lagranxiana" title="Mecánica lagranxiana – galisisk" lang="gl" hreflang="gl" data-title="Mecánica lagranxiana" data-language-autonym="Galego" data-language-local-name="galisisk" 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/%EB%9D%BC%EA%B7%B8%EB%9E%91%EC%A3%BC_%EC%97%AD%ED%95%99" title="라그랑주 역학 – koreansk" lang="ko" hreflang="ko" data-title="라그랑주 역학" data-language-autonym="한국어" data-language-local-name="koreansk" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B2%E0%A4%BE%E0%A4%97%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%82%E0%A4%9C%E0%A5%80%E0%A4%AF_%E0%A4%AF%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80" title="लाग्रांजीय यांत्रिकी – hindi" lang="hi" hreflang="hi" data-title="लाग्रांजीय यांत्रिकी" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Mekanika_Lagrangian" title="Mekanika Lagrangian – indonesisk" lang="id" hreflang="id" data-title="Mekanika Lagrangian" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesisk" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Meccanica_lagrangiana" title="Meccanica lagrangiana – italiensk" lang="it" hreflang="it" data-title="Meccanica lagrangiana" data-language-autonym="Italiano" data-language-local-name="italiensk" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%9B%D7%A0%D7%99%D7%A7%D7%94_%D7%90%D7%A0%D7%9C%D7%99%D7%98%D7%99%D7%AA" title="מכניקה אנליטית – hebraisk" lang="he" hreflang="he" data-title="מכניקה אנליטית" data-language-autonym="עברית" data-language-local-name="hebraisk" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Mekanika_Lagrangian" title="Mekanika Lagrangian – javanesisk" lang="jv" hreflang="jv" data-title="Mekanika Lagrangian" data-language-autonym="Jawa" data-language-local-name="javanesisk" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F%D1%81%D1%8B" title="Лагранж функциясы – kasakhisk" lang="kk" hreflang="kk" data-title="Лагранж функциясы" data-language-autonym="Қазақша" data-language-local-name="kasakhisk" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B2%E0%B4%97%E0%B5%8D%E0%B4%B0%E0%B4%BE%E0%B4%9E%E0%B5%8D%E0%B4%9A%E0%B4%BF%E0%B4%AF%E0%B5%BB_%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="ലഗ്രാഞ്ചിയൻ ബലതന്ത്രം – malayalam" lang="ml" hreflang="ml" data-title="ലഗ്രാഞ്ചിയൻ ബലതന്ത്രം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lagrangiaanse_mechanica" title="Lagrangiaanse mechanica – nederlandsk" lang="nl" hreflang="nl" data-title="Lagrangiaanse mechanica" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" 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/%E3%83%A9%E3%82%B0%E3%83%A9%E3%83%B3%E3%82%B8%E3%83%A5%E5%8A%9B%E5%AD%A6" title="ラグランジュ力学 – japansk" lang="ja" hreflang="ja" data-title="ラグランジュ力学" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Mechanika_Lagrange%E2%80%99a" title="Mechanika Lagrange’a – polsk" lang="pl" hreflang="pl" data-title="Mechanika Lagrange’a" data-language-autonym="Polski" data-language-local-name="polsk" 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_de_Lagrange" title="Mecânica de Lagrange – portugisisk" lang="pt" hreflang="pt" data-title="Mecânica de Lagrange" data-language-autonym="Português" data-language-local-name="portugisisk" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Mecanic%C4%83_lagrangian%C4%83" title="Mecanică lagrangiană – rumensk" lang="ro" hreflang="ro" data-title="Mecanică lagrangiană" data-language-autonym="Română" data-language-local-name="rumensk" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%B5%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Лагранжева механика – russisk" lang="ru" hreflang="ru" data-title="Лагранжева механика" data-language-autonym="Русский" data-language-local-name="russisk" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Mekanika_e_Lagranzhit" title="Mekanika e Lagranzhit – albansk" lang="sq" hreflang="sq" data-title="Mekanika e Lagranzhit" data-language-autonym="Shqip" data-language-local-name="albansk" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Lagrangeeva_formulacija_gibalnih_ena%C4%8Db" title="Lagrangeeva formulacija gibalnih enačb – slovensk" lang="sl" hreflang="sl" data-title="Lagrangeeva formulacija gibalnih enačb" data-language-autonym="Slovenščina" data-language-local-name="slovensk" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">flytt til sidefeltet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skjul</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Fra Wikipedia, den frie encyklopedi</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="nb" dir="ltr"><p><b>Lagrange-mekanikk</b> er en mer generell formulering av <a href="/wiki/Klassisk_mekanikk" title="Klassisk mekanikk">klassisk mekanikk</a> enn den som ble innført av <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>. I stedet for en formulering av bevegelseslovene for et mekanisk system uttrykt ved <a href="/wiki/Akselerasjon" title="Akselerasjon">akselerasjonen</a> forårsaket av alle krefter som virker på det, viste den franske fysiker og matematiker <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a> i <a href="/wiki/1788" title="1788">1788</a> ut fra <a href="/w/index.php?title=D%27Alemberts_prinsipp&action=edit&redlink=1" class="new" title="D'Alemberts prinsipp (ikke skrevet ennå)">d'Alemberts prinsipp</a> at de kan utledes fra en skalar funksjon av systemets uavhengige variable og deres tidsderiverte. Denne funksjonen kalles i dag for <b>Lagrange-funksjonen</b>. For et system med en veldefinert <a href="/wiki/Kinetisk_energi" title="Kinetisk energi">kinetisk energi</a> <i>T</i> og <a href="/wiki/Potensiell_energi" title="Potensiell energi">potensiell energi</a> <i>V</i> er Lagrange-funksjonen <i>L = T - V</i>. </p><p>Et generelt system kan beskrives ved <i>N</i> uavhengige variable eller <i>generelle</i> koordinater <i>q</i> = <i>(q<sub>1</sub>, q<sub>2</sub>, ... , q<sub>N</sub>)</i>. Disse behøver ikke å være komponenter av forskjellige posisjonsvektorer, men kan for eksempel oppstå ved bruk av ikke-kartesiske koordinatsystem. Lagrange-funksjonen er da av formen </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 L=L(q,{\dot {q}},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=L(q,{\dot {q}},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5688f7aefed66333264e4c29a78a1ddd416be66c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.428ex; height:2.843ex;" alt="{\displaystyle L=L(q,{\dot {q}},t)}"></span></dd></dl> <p>hvor de tidsderiverte er <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}}={\frac {dq}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>q</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {q}}={\frac {dq}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f080c7994b6c59b5f2ce39909c43d798c607dd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.597ex; height:5.509ex;" alt="{\displaystyle {\dot {q}}={\frac {dq}{dt}}}"></span>. Den klassiske bevegelsen er nå gitt som løsningen av <a href="/wiki/Differensialligning" title="Differensialligning">differensialligningen</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 {d \over dt}\left({\partial L \over \partial {\dot {q}}_{n}}\right)-{\partial L \over \partial q_{n}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dt}\left({\partial L \over \partial {\dot {q}}_{n}}\right)-{\partial L \over \partial q_{n}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73ee4d83ebc047b09591917ca34ccea4eb2985d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.961ex; height:6.176ex;" alt="{\displaystyle {d \over dt}\left({\partial L \over \partial {\dot {q}}_{n}}\right)-{\partial L \over \partial q_{n}}=0}"></span></dd></dl> <p>som kalles <b>Euler-Lagrange-ligningen</b>. Den var tidligere blitt utledet av <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> og <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Lagrange</a> i forbindelse med løsning av forskjellige optimaliseringsproblem ved bruk av <a href="/wiki/Variasjonsregning" title="Variasjonsregning">variasjonsregning</a>. </p><p>Men det var den irske fysiker og matematiker <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> som på midten 18-hundreårstallet viste at den samme ligningen kan utledes fra et <a href="/wiki/Virkningsprinsipp" title="Virkningsprinsipp">virkningsprinsipp</a>. Han innførte en ny definisjon av begrepet mekanisk virkning. For en bevegelse av systemet fra en gitt tilstand <i>A</i> til en senere tilstand <i>B</i> er virkningen definert ved integralet </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 S=\int _{A}^{B}\!dtL(q,{\dot {q}},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <mi>d</mi> <mi>t</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\int _{A}^{B}\!dtL(q,{\dot {q}},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cba2d9ca6fc06a2f465a0a8938465716ee770630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.343ex; height:6.176ex;" alt="{\displaystyle S=\int _{A}^{B}\!dtL(q,{\dot {q}},t)}"></span></dd></dl> <p>hvor <i>L</i> er Lagrange-funksjonen for systemet. Den klassiske bevegelsen følger en bane hvis virkning skal ha et ekstremum, vanligvis et minimum. Under små variasjoner δ<i>q</i> rundt denne klassiske banen er virkningen derfor <i>stasjonær</i>, det vil si at den resulterende variasjon </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 S=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>S</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta S=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be02bab96c7b908ef7d379d67f1d3aca2fdfde29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.456ex; height:2.343ex;" alt="{\displaystyle \delta S=0.}"></span></dd></dl> <p>Ved bruk av standard <a href="/wiki/Variasjonsregning" title="Variasjonsregning">variasjonsregning</a> finner man da at den klassiske bevegelsen må oppfylle Euler-Lagrange-ligningen. Dette virkningsprinsippet kalles vanligvis i dag for <a href="/wiki/Hamiltons_virkningsprinsipp" title="Hamiltons virkningsprinsipp">Hamiltons virkningsprinsipp</a> og spiller en fundamentalt viktig rolle i moderne fysikk. Ofte blir det omtalt som <b>prinsippet om minste virkning</b> selv om det navnet noen ganger kan være misvisende. </p><p>Lagrange-mekanikk kan også brukes til beskrivelse av kontinuerlige systemer og <a href="/wiki/Felt" class="mw-disambig" title="Felt">felt</a>. Lagrange-funksjonen er da gitt som et volumintegral over en <i>Lagrange-tetthet</i>. Moderne <a href="/wiki/Kvantefeltteori" title="Kvantefeltteori">kvantefeltteorier</a> formuleres alle på denne måten. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Generelle_koordinater">Generelle koordinater</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=1" title="Rediger avsnitt: Generelle koordinater" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=1" title="Rediger kildekoden til seksjonen Generelle koordinater"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Newtons_lover" class="mw-redirect" title="Newtons lover">Newtons andre lov</a> er vanligvis formulert i et <a href="/wiki/Kartesisk_koordinatsystem" title="Kartesisk koordinatsystem">kartesisk koordinatsystem</a> hvor hver partikkel er angitt ved en posisjonsvektor <b>r</b> = <b>r</b>(<i>t</i>) og en tilsvarende hastighet <b>v</b> = d<b>r</b>/d<i>t</i>, hver med tre komponenter. Den deriverte av hastigheten er <a href="/wiki/Akselerasjon" title="Akselerasjon">akselerasjonen</a> <b>a</b> = d<b>v</b>/d<i>t</i> som er proporsjonal med kraften <b>F</b> som virker på denne partikkelen. Bevegelsesligningene <b>F</b> = <i>m</i><b>a</b> blir da andre ordens <a href="/wiki/Differensialligning" title="Differensialligning">differensialligninger</a> for hver av kompontene for posisjonsvektoren <b>r</b>. I utgangspunktet er derfor bevegelsen til en partikkel karakterisert ved tre variable eller frihetsgrader. Men disse variable er ikke alltid uavhengige av hverandre. </p><p>Ofte kan det være vanskelig å utlede bevegelsesligningene fordi mange av kreftene som inngår, er <i>føringskrefter</i> som må innføres for at bevegelsen skal oppfylle gitte krav. Dette kan for eksempel være at partikkelen skal bevege seg på en bestemt flate eller i en gitt avstand fra et fiksert punkt. Det ville være enklere å kun benytte de variable som trenges for å angi tillatte posisjoner slik at føringskreftene ikke lenger inngår i problemet. Dette kalles <i>generaliserte koordinater</i> og deres antall er det effektive antall av frihetsgrader.<sup id="cite_ref-Goldstein_1-0" class="reference"><a href="#cite_note-Goldstein-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>For eksempel vil enden til en <a href="/wiki/Pendel" title="Pendel">pendel</a> med lengde <i>a</i> som svinger i <i>xy</i>-planet ha en posisjon <b>r</b>(t) = (<i>x(t),y(t</i>) = <i>a</i> (sin<i>θ</i>,cos<i>θ</i>) hvor vinkelen <i>θ</i> er utslaget fra <i>y</i>-aksen. Dette er derfor den generaliserte koordinaten for pendelen som dermed har bare en frihetsgrad. En partikkel som beveger seg på en gitt flate, vil derimot ha to frihetsgrader som er dens koordinater på flaten. </p> <div class="mw-heading mw-heading3"><h3 id="Kinematiske_relasjoner">Kinematiske relasjoner</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=2" title="Rediger avsnitt: Kinematiske relasjoner" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=2" title="Rediger kildekoden til seksjonen Kinematiske relasjoner"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I alminnelighet tenker vi oss at systemet kan beskrives ved <i>N</i> generaliserte koordinater <i>q</i> = <i>(q<sub>1</sub>, q<sub>2</sub>, ... q<sub>N</sub></i>). Betrakter vi partikkel angitt ved indeks <i>a</i>, vil den da ha posisjonsvektor </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 {r} _{a}=\mathbf {r} _{a}(q_{1},q_{2},\cdots ,q_{N},t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{a}=\mathbf {r} _{a}(q_{1},q_{2},\cdots ,q_{N},t).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0918dcf9911f40b37082dc9071fd4cebd5e6f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.959ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} _{a}=\mathbf {r} _{a}(q_{1},q_{2},\cdots ,q_{N},t).}"></span></dd></dl> <p>når man tillater at denne sammenhengen i alminnelighet også kan være eksplisitt avhengig av tiden <i>t</i>. Hastigheten til partikkelen kan derfor skrives som </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 {\dot {\mathbf {r} }}_{a}={d\mathbf {r} _{a} \over dt}=\sum _{n}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\dot {q}}_{n}+{\partial \mathbf {r} _{a} \over \partial t}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {r} }}_{a}={d\mathbf {r} _{a} \over dt}=\sum _{n}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\dot {q}}_{n}+{\partial \mathbf {r} _{a} \over \partial t}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/712e897f1765f17f28a57e7aec5c355ed1e09558" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.31ex; height:6.509ex;" alt="{\displaystyle {\dot {\mathbf {r} }}_{a}={d\mathbf {r} _{a} \over dt}=\sum _{n}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\dot {q}}_{n}+{\partial \mathbf {r} _{a} \over \partial t}.}"></span></dd></dl> <p>Herfra finner man nå at </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 {\partial {\dot {\mathbf {r} }}_{a} \over \partial {\dot {q}}_{n}}={\partial \mathbf {r} _{a} \over \partial q_{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial {\dot {\mathbf {r} }}_{a} \over \partial {\dot {q}}_{n}}={\partial \mathbf {r} _{a} \over \partial q_{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b36074691458efc4e78b9ddc91f9d748e48a05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.905ex; height:6.009ex;" alt="{\displaystyle {\partial {\dot {\mathbf {r} }}_{a} \over \partial {\dot {q}}_{n}}={\partial \mathbf {r} _{a} \over \partial q_{n}}.}"></span></dd></dl> <p>Prikkene fra tidsderivasjonene ser ut til å ha kansellert hverandre. Dette resultatet benyttes nå til å omskrive det skalare produktet </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 {\dot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}={\dot {\mathbf {r} }}_{a}\cdot {\partial {\dot {\mathbf {r} }}_{a} \over \partial {\dot {q}}_{n}}={1 \over 2}{\partial {\dot {\mathbf {r} }}_{a}^{2} \over \partial {\dot {q}}_{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}={\dot {\mathbf {r} }}_{a}\cdot {\partial {\dot {\mathbf {r} }}_{a} \over \partial {\dot {q}}_{n}}={1 \over 2}{\partial {\dot {\mathbf {r} }}_{a}^{2} \over \partial {\dot {q}}_{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3da5451517a088e8d2f41ea035dcdabc3a6dd818" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.639ex; height:6.676ex;" alt="{\displaystyle {\dot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}={\dot {\mathbf {r} }}_{a}\cdot {\partial {\dot {\mathbf {r} }}_{a} \over \partial {\dot {q}}_{n}}={1 \over 2}{\partial {\dot {\mathbf {r} }}_{a}^{2} \over \partial {\dot {q}}_{n}}.}"></span></dd></dl> <p>Via det tidsderiverte uttrykket </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 {d \over dt}\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}={d \over dt}{\Big [}{\partial \over \partial {\dot {q}}_{n}}{\Big (}\sum _{a}{1 \over 2}m_{a}{\dot {\mathbf {r} }}_{a}^{2}{\Big )}{\Big ]}={d \over dt}{\Big (}{\partial T \over \partial {\dot {q}}_{n}}{\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dt}\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}={d \over dt}{\Big [}{\partial \over \partial {\dot {q}}_{n}}{\Big (}\sum _{a}{1 \over 2}m_{a}{\dot {\mathbf {r} }}_{a}^{2}{\Big )}{\Big ]}={d \over dt}{\Big (}{\partial T \over \partial {\dot {q}}_{n}}{\Big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2872bb630687decc5971da375a8e6657136af506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:59.276ex; height:6.509ex;" alt="{\displaystyle {d \over dt}\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}={d \over dt}{\Big [}{\partial \over \partial {\dot {q}}_{n}}{\Big (}\sum _{a}{1 \over 2}m_{a}{\dot {\mathbf {r} }}_{a}^{2}{\Big )}{\Big ]}={d \over dt}{\Big (}{\partial T \over \partial {\dot {q}}_{n}}{\Big )}}"></span></dd></dl> <p>har man dermed etablert en forbindelse med den kinetiske energien, </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 T={1 \over 2}\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={1 \over 2}\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a2237f66cefa9aa7499aed4f73aaa7418483ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.916ex; height:6.343ex;" alt="{\displaystyle T={1 \over 2}\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}^{2},}"></span></dd></dl> <p>til alle partiklene i systemet.<sup id="cite_ref-Goldstein_1-1" class="reference"><a href="#cite_note-Goldstein-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>En lignende relasjon kan også utledes fra den deriverte </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 {\partial {\dot {\mathbf {r} }}_{a} \over \partial q_{n}}={\partial \over \partial q_{n}}{\Big (}\sum _{m}{\partial \mathbf {r} _{a} \over \partial q_{m}}{\dot {q}}_{m}+{\partial \mathbf {r} _{a} \over \partial t}{\Big )}=\sum _{m}{\partial ^{2}\mathbf {r} _{a} \over \partial q_{m}\partial q_{n}}{\dot {q}}_{m}+{\partial ^{2}\mathbf {r} _{a} \over \partial q_{n}\partial t}={d \over dt}{\Big (}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial {\dot {\mathbf {r} }}_{a} \over \partial q_{n}}={\partial \over \partial q_{n}}{\Big (}\sum _{m}{\partial \mathbf {r} _{a} \over \partial q_{m}}{\dot {q}}_{m}+{\partial \mathbf {r} _{a} \over \partial t}{\Big )}=\sum _{m}{\partial ^{2}\mathbf {r} _{a} \over \partial q_{m}\partial q_{n}}{\dot {q}}_{m}+{\partial ^{2}\mathbf {r} _{a} \over \partial q_{n}\partial t}={d \over dt}{\Big (}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd6fceab51d8bf96be11cf1971c4af727002aab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:75.511ex; height:6.843ex;" alt="{\displaystyle {\partial {\dot {\mathbf {r} }}_{a} \over \partial q_{n}}={\partial \over \partial q_{n}}{\Big (}\sum _{m}{\partial \mathbf {r} _{a} \over \partial q_{m}}{\dot {q}}_{m}+{\partial \mathbf {r} _{a} \over \partial t}{\Big )}=\sum _{m}{\partial ^{2}\mathbf {r} _{a} \over \partial q_{m}\partial q_{n}}{\dot {q}}_{m}+{\partial ^{2}\mathbf {r} _{a} \over \partial q_{n}\partial t}={d \over dt}{\Big (}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}.}"></span></dd></dl> <p>Herfra følger nå på tilsvarende måte som ovenfor at </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 _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {d \over dt}{\Big (}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}=\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {\partial {\dot {\mathbf {r} }}_{a} \over \partial q_{n}}={\partial T \over \partial q_{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {d \over dt}{\Big (}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}=\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {\partial {\dot {\mathbf {r} }}_{a} \over \partial q_{n}}={\partial T \over \partial q_{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82f5ee3ec9021534b3f20df68d61c2c8579402da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.404ex; height:6.509ex;" alt="{\displaystyle \sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {d \over dt}{\Big (}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}=\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {\partial {\dot {\mathbf {r} }}_{a} \over \partial q_{n}}={\partial T \over \partial q_{n}}.}"></span></dd></dl> <p>Disse matematiske relasjonene mellom de kinematiske variable kan nå benyttes i den dynamiske beskrivelsen av systemet. </p> <div class="mw-heading mw-heading2"><h2 id="D'Alemberts_prinsipp"><span id="D.27Alemberts_prinsipp"></span>D'Alemberts prinsipp</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=3" title="Rediger avsnitt: D'Alemberts prinsipp" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=3" title="Rediger kildekoden til seksjonen D'Alemberts prinsipp"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For et statisk system sier <a href="/w/index.php?title=D%27Alemberts_prinsipp&action=edit&redlink=1" class="new" title="D'Alemberts prinsipp (ikke skrevet ennå)">d'Alemberts prinsipp</a> at under en meget liten og tenkt forskyving δ<b>r</b><sub><i>a</i></sub> av partikkel <i>a</i> som påvirkes av den totale kraften <b>F</b><sub><i>a</i></sub> , skal det resulterende arbeid være null. På matematisk form kan det skrives som </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 _{a}\mathbf {F} _{a}\cdot \delta \mathbf {r} _{a}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{a}\mathbf {F} _{a}\cdot \delta \mathbf {r} _{a}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde6ef818ddb98aeec3d8b7b8720df9ef22b8c43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.366ex; height:5.509ex;" alt="{\displaystyle \sum _{a}\mathbf {F} _{a}\cdot \delta \mathbf {r} _{a}=0.}"></span></dd></dl> <p>Den infinitesemale forskyvingen δ<b>r</b><sub><i>a</i></sub>, som må oppfylle føringsbetingelsene, sies å være <b>virtuell</b> da den skal foregå ved konstant tid. Ingen virkelig forflytning kan i foregå uten at den tar litt tid. Derfor er den man betrakter her, virtuell. Ved bruk av generaliserte koordinater, kan den da skrives som </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 \mathbf {r} _{a}=\sum _{n}{\partial \mathbf {r} _{a} \over \partial q_{n}}\delta q_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \mathbf {r} _{a}=\sum _{n}{\partial \mathbf {r} _{a} \over \partial q_{n}}\delta q_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1666c1f4def79295352b01225ff597a732a2e9e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.454ex; height:6.509ex;" alt="{\displaystyle \delta \mathbf {r} _{a}=\sum _{n}{\partial \mathbf {r} _{a} \over \partial q_{n}}\delta q_{n},}"></span></dd></dl> <p>hvor det på høyre side i alminnelighet skulle ha vært et ledd (∂<b>r</b><sub><i>a</i></sub>/∂<i>t</i>)δ<i>t</i> . </p><p>Dette prinsippet kan utvides til å også gjelde for et dynamisk system ved å anta at <a href="/wiki/Akselerasjon" title="Akselerasjon">akselerasjonen</a> <b>a</b><sub><i>a</i></sub> til denne partikkelen i systemet forårsaker en virkende kraft på samme måte som for eksempel at <a href="/wiki/Sentrifugalkraft" title="Sentrifugalkraft">sentrifugalkraften</a> kan beskrives som en virkelig kraft. Kombinert med <a href="/wiki/Newtons_lover" class="mw-redirect" title="Newtons lover">Newtons andre lov</a> kan man da skrive d'Alemberts dynamiske prinsipp på den matematiske formen </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 _{a}(\mathbf {F} _{a}-m_{a}\mathbf {a} _{a})\cdot \delta \mathbf {r} _{a}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{a}(\mathbf {F} _{a}-m_{a}\mathbf {a} _{a})\cdot \delta \mathbf {r} _{a}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd55b16c13ebac34bff9528cc4b39ebd6b13ccbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.172ex; height:5.509ex;" alt="{\displaystyle \sum _{a}(\mathbf {F} _{a}-m_{a}\mathbf {a} _{a})\cdot \delta \mathbf {r} _{a}=0.}"></span></dd></dl> <p>Den totale kraften <b>F</b><sub><i>a</i></sub> som virker på partikkel <i>a</i>, består av en ytre kraft <b>F</b><sub><i>a</i></sub><sup><i>e</i></sup> og en indre føringskraft <b>F</b><sub><i>a</i></sub><sup><i>i</i></sup> slik at totalkraften er <b>F</b><sub><i>a</i></sub> = <b>F</b><sub><i>a</i></sub><sup><i>e</i></sup> + <b>F</b><sub><i>a</i></sub><sup><i>i</i></sup> . Føringskreftene kan ikke utføre noe arbeid slik at </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 _{a}\mathbf {F} _{a}^{i}\cdot \delta \mathbf {r} _{a}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{a}\mathbf {F} _{a}^{i}\cdot \delta \mathbf {r} _{a}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a497b346a6fe0921b2555623b4137022cb106659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.366ex; height:5.509ex;" alt="{\displaystyle \sum _{a}\mathbf {F} _{a}^{i}\cdot \delta \mathbf {r} _{a}=0.}"></span></dd></dl> <p>Dermed inneholder d'Alemberts dynamiske prinsipp kun de ytre kreftene som virker på systemet, </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 _{a}(\mathbf {F} _{a}^{e}-m_{a}\mathbf {a} _{a})\cdot \delta \mathbf {r} _{a}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{a}(\mathbf {F} _{a}^{e}-m_{a}\mathbf {a} _{a})\cdot \delta \mathbf {r} _{a}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f04950238a30849b815c68e860cd33ee22766d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.525ex; height:5.509ex;" alt="{\displaystyle \sum _{a}(\mathbf {F} _{a}^{e}-m_{a}\mathbf {a} _{a})\cdot \delta \mathbf {r} _{a}=0}"></span>,</dd></dl> <p>slik at problemet er blitt betydelig forenklet. Her kan vi nå sette inn uttrykket for den virtuelle forskyvningen uttrykt i generelle koordinater, </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 _{a,n}(\mathbf {F} _{a}^{e}-m_{a}{\ddot {\mathbf {r} }}_{a})\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}\delta q_{n}=\sum _{n}{\Big (}Q_{n}-\sum _{a}m_{a}{\ddot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}\delta q_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>n</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mi>δ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{a,n}(\mathbf {F} _{a}^{e}-m_{a}{\ddot {\mathbf {r} }}_{a})\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}\delta q_{n}=\sum _{n}{\Big (}Q_{n}-\sum _{a}m_{a}{\ddot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}\delta q_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf1fbef815998a7b93e558cdf719ec4022fd844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.291ex; height:6.676ex;" alt="{\displaystyle \sum _{a,n}(\mathbf {F} _{a}^{e}-m_{a}{\ddot {\mathbf {r} }}_{a})\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}\delta q_{n}=\sum _{n}{\Big (}Q_{n}-\sum _{a}m_{a}{\ddot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}\delta q_{n},}"></span></dd></dl> <p>hvor vi har innført </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 Q_{n}=\sum _{a}\mathbf {F} _{a}^{e}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n}=\sum _{a}\mathbf {F} _{a}^{e}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc7e53949af45096ed46625eedaa5d5028ed52a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.418ex; height:6.509ex;" alt="{\displaystyle Q_{n}=\sum _{a}\mathbf {F} _{a}^{e}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}},}"></span></dd></dl> <p>som kalles en <b>generell kraft</b>. Dette kommer frem ved å betrakte det spesielle tilfellet at de ytre kreftene kan avledes av et potensial <i>V</i> = <i>V</i>(<b>r</b><sub>1</sub>, <b>r</b><sub>2</sub>, ....). Da er </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} _{a}^{e}=-{\boldsymbol {\nabla }}_{a}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{a}^{e}=-{\boldsymbol {\nabla }}_{a}V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63a610b529c7aff1faa87ccdebd4643856cc311f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.806ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} _{a}^{e}=-{\boldsymbol {\nabla }}_{a}V}"></span>,</dd></dl> <p>slik at den generelle kraften blir </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 Q_{n}=\sum _{a}{\boldsymbol {\nabla }}_{a}V\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}=-{\partial V \over \partial q_{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mi>V</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n}=\sum _{a}{\boldsymbol {\nabla }}_{a}V\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}=-{\partial V \over \partial q_{n}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ce977121679d13c2112c069ffdbb43cbeabe94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.065ex; height:6.509ex;" alt="{\displaystyle Q_{n}=\sum _{a}{\boldsymbol {\nabla }}_{a}V\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}=-{\partial V \over \partial q_{n}},}"></span></dd></dl> <p>som man ville forvente ved innføring av generelle koordinater.<sup id="cite_ref-Goldstein_1-2" class="reference"><a href="#cite_note-Goldstein-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Denne kraften opptrer sammen med akselerasjonsleddet som vi kan splitte i to ved å skrive </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 _{a}m_{a}{\ddot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}={d \over dt}{\Big (}\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}-\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {d \over dt}{\Big (}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{a}m_{a}{\ddot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}={d \over dt}{\Big (}\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}-\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {d \over dt}{\Big (}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfafd8cd83d465ed27b3655152addc10b91889f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:64.019ex; height:6.509ex;" alt="{\displaystyle \sum _{a}m_{a}{\ddot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}={d \over dt}{\Big (}\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}-\sum _{a}m_{a}{\dot {\mathbf {r} }}_{a}\cdot {d \over dt}{\Big (}{\partial \mathbf {r} _{a} \over \partial q_{n}}{\Big )}.}"></span></dd></dl> <p>Begge termene på høyre side har vi allerede vist at kan uttrykkes ved deriverte av den kinetiske energien <i>T</i>. Setter vi inn de resultatene her, vil hvert ledd i summen inneholde den generelle forskyvningen δ<i>q<sub>n</sub></i>. Da alle disse er vilkårlige og uavhengige av hverandre, må derfor hver prefaktor til δ<i>q<sub>n</sub></i> være null. Det betyr at den ene ligningen man startet ut fra, splittes opp i en ligning </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 {d \over dt}{\Big (}{\partial T \over \partial {\dot {q}}_{n}}{\Big )}-{\partial T \over \partial q_{n}}=Q_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dt}{\Big (}{\partial T \over \partial {\dot {q}}_{n}}{\Big )}-{\partial T \over \partial q_{n}}=Q_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ab553236218c5d46c34ce57c389a374c9aafd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.822ex; height:6.009ex;" alt="{\displaystyle {d \over dt}{\Big (}{\partial T \over \partial {\dot {q}}_{n}}{\Big )}-{\partial T \over \partial q_{n}}=Q_{n}}"></span></dd></dl> <p>for hver generell koordinat. Dermed har man funnet de nye bevegelsesligningene som følger fra d'Alemberts prinsipp. </p> <div class="mw-heading mw-heading2"><h2 id="Lagrange-funksjonen">Lagrange-funksjonen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=4" title="Rediger avsnitt: Lagrange-funksjonen" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=4" title="Rediger kildekoden til seksjonen Lagrange-funksjonen"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bevegelsesligningene kan gjøres enklere ved å uttrykke de generelle kreftene ved den generelle, potensielle energien <i>V</i> som ble omtalt over. Er denne uavhengig av hastighetene til partiklene, finner vi ved å innføre at <i>Q<sub>n</sub></i> = - <i>∂V/∂q<sub>n</sub></i>, at bevegelsesligningene kan skrives som </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 {d \over dt}{\Big (}{\partial L \over \partial {\dot {q}}_{n}}{\Big )}-{\partial L \over \partial q_{n}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dt}{\Big (}{\partial L \over \partial {\dot {q}}_{n}}{\Big )}-{\partial L \over \partial q_{n}}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a3607a65bb3256381d2122e24dd6c9689a393c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.575ex; height:6.009ex;" alt="{\displaystyle {d \over dt}{\Big (}{\partial L \over \partial {\dot {q}}_{n}}{\Big )}-{\partial L \over \partial q_{n}}=0,}"></span></dd></dl> <p>hvor <b>Lagrange-funksjonen</b> er <i>L = T - V</i>. Dette er Euler-Lagrange-ligningen som først oppstod tidligere i forbindelse med løsningen av forskjellige <a href="/wiki/Variasjonsregning" title="Variasjonsregning">variasjonsproblem</a>. Den vil inneholde dobbeltderiverte av de variable slik at den er en annenordens <a href="/wiki/Differensialligning" title="Differensialligning">differensialligning</a>. En fullstendig løsning krever derfor to grensebetingelser for hver variabel. Vanligvis blir de angitt som initialbetingelser ved tiden <i>t = 0</i> og kan være posisjon og hastighet ved dette tidspunktet.<sup id="cite_ref-Goldstine_2-0" class="reference"><a href="#cite_note-Goldstine-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Som et enkelt eksempel kan man betrakte en <a href="/wiki/Harmonisk_oscillator" title="Harmonisk oscillator">harmonisk oscillator</a> med utslag <i>q</i> og kraftkonstant <i>k</i>. Den potensielle energien er derfor <i>V =(1/2) kq<sup>2</sup></i> slik at Lagrange-funksjonen blir </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 L={1 \over 2}m{\dot {q}}^{2}-{1 \over 2}kq^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={1 \over 2}m{\dot {q}}^{2}-{1 \over 2}kq^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e723f1afb7b4674602e1531bcd1d08662e2ce87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.982ex; height:5.176ex;" alt="{\displaystyle L={1 \over 2}m{\dot {q}}^{2}-{1 \over 2}kq^{2}.}"></span></dd></dl> <p>Herfra følger at den deriverte </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 {\partial L \over \partial {\dot {q}}}=m{\dot {q}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial L \over \partial {\dot {q}}}=m{\dot {q}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffee3ee2ab3e937b548bc20e03f170fd2dcb10b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.9ex; height:5.843ex;" alt="{\displaystyle {\partial L \over \partial {\dot {q}}}=m{\dot {q}},}"></span></dd></dl> <p>mens <i>∂ L/∂ q</i> = -<i>kq</i> slik at bevegelsesligningen blir </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 m{d^{2}q \over dt^{2}}+kq^{2}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>q</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>k</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{d^{2}q \over dt^{2}}+kq^{2}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1db9dd227674fed7228ed030197bf287781200a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.311ex; height:6.009ex;" alt="{\displaystyle m{d^{2}q \over dt^{2}}+kq^{2}=0.}"></span></dd></dl> <p>Den generelle løsningen kan skrives på formen <i>q(t) = a</i> sin(ω<i>t</i> + φ) med vinkelfrekvens ω = <i>√(k/m)</i>. Amplituden <i>a</i> og fasen φ er integrasjonskonstanter som må bestemmes ut fra gitte grensebetingelser. </p><p>For en partikkel med masse <i>m</i> er Lagrange-funksjonen mer generelt </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 L={1 \over 2}m{\dot {\mathbf {r} }}^{2}-V(\mathbf {r} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={1 \over 2}m{\dot {\mathbf {r} }}^{2}-V(\mathbf {r} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b0aec776a4d24c38be4a9818932d98931b5723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.123ex; height:5.176ex;" alt="{\displaystyle L={1 \over 2}m{\dot {\mathbf {r} }}^{2}-V(\mathbf {r} ),}"></span></dd></dl> <p>hvor <i>V = V(<b>r</b>)</i> er den potensielle energien i posisjonen <b>r</b> = (<i>x,y,z</i>). Nå er </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 {\partial L \over \partial {\dot {\mathbf {r} }}}=m{\dot {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial L \over \partial {\dot {\mathbf {r} }}}=m{\dot {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3793e0b929d474302d039573a3b3b9997363e888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.038ex; height:5.509ex;" alt="{\displaystyle {\partial L \over \partial {\dot {\mathbf {r} }}}=m{\dot {\mathbf {r} }}}"></span></dd></dl> <p>og <i>∂ L/∂ <b>r</b></i> = - <b>∇</b><i>V</i> som sammen gir bevegelsesligningen </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 m{d\mathbf {v} \over dt}=-{\boldsymbol {\nabla }}V(\mathbf {r} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{d\mathbf {v} \over dt}=-{\boldsymbol {\nabla }}V(\mathbf {r} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5b0640cb293b0f5e6b8576db08af63059fd47b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.982ex; height:5.509ex;" alt="{\displaystyle m{d\mathbf {v} \over dt}=-{\boldsymbol {\nabla }}V(\mathbf {r} ),}"></span></dd></dl> <p>hvor <b>v</b> = d<b>r</b>/d<i>t</i> er hastigheten til partikkelen. Dette resultatet kjenner man igjen som <a href="/wiki/Newtons_bevegelseslover" title="Newtons bevegelseslover"> Newtons andre ligning</a> hvor <b>F</b> = - <b>∇</b><i>V</i> er kraften som virker på partikkelen. Det er ikke så overraskende da den jo er utgangspunktet for det hele. </p><p>Euler-Lagrange-ligningen gjelder også i det mer generelle tilfellet at den potensielle energien <i>V</i> er avhengig av hastighetene til partiklene på en slik måte at den generelle kraften kan skrives som </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 Q_{n}=-{\partial V \over \partial q_{n}}+{d \over dt}{\Big (}{\partial V \over \partial {\dot {q}}_{n}}{\Big )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n}=-{\partial V \over \partial q_{n}}+{d \over dt}{\Big (}{\partial V \over \partial {\dot {q}}_{n}}{\Big )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/703ab5653d87567cc01b80c9488bf9d40253da1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.277ex; height:6.009ex;" alt="{\displaystyle Q_{n}=-{\partial V \over \partial q_{n}}+{d \over dt}{\Big (}{\partial V \over \partial {\dot {q}}_{n}}{\Big )}.}"></span></dd></dl> <p>Det er tilfellet for en partikkel i et <a href="/wiki/Magnetisk_felt" class="mw-redirect" title="Magnetisk felt">magnetisk felt</a> <b>B</b> = <b>∇ </b>× <b>A</b> hvor <b>A</b> = <b>A</b>(<b>r</b>,<i>t</i>) er vektorpotensialet. For en partikkel med ladning <i>q</i> er da den potensielle energien <span class="nowrap"><i>V</i> = <i>- q</i> <b>v</b>⋅<b>A</b></span> når den har hastighet <span class="nowrap"><b>v</b> = d<b>r</b>/d<i>t</i></span>. Fra Lagrange-funksjonen </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 L={1 \over 2}m\mathbf {v} ^{2}+q\mathbf {v} \cdot \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={1 \over 2}m\mathbf {v} ^{2}+q\mathbf {v} \cdot \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd28b561891cc75bd7c988fb4f6951b9e1076379" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.205ex; height:5.176ex;" alt="{\displaystyle L={1 \over 2}m\mathbf {v} ^{2}+q\mathbf {v} \cdot \mathbf {A} }"></span></dd></dl> <p>følger da bevegelsesligningen </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 m{d\mathbf {v} \over dt}=q\mathbf {v} \times \mathbf {B} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{d\mathbf {v} \over dt}=q\mathbf {v} \times \mathbf {B} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38ec2a5a6835071d0286ac96ea89b7ca60c2637b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.471ex; height:5.509ex;" alt="{\displaystyle m{d\mathbf {v} \over dt}=q\mathbf {v} \times \mathbf {B} .}"></span></dd></dl> <p>På høyre side opptrer nå <a href="/wiki/Lorentz-kraft" title="Lorentz-kraft">Lorentz-kraften</a>. Ut fra definisjonen av <a href="/wiki/Vektor_(matematikk)" title="Vektor (matematikk)">kryssproduktet</a> mellom vektorer, virker den normalt på hastigheten. Den vil derfor ikke forandre størrelsen til hastigheten, men bare dens retning.<sup id="cite_ref-YF_3-0" class="reference"><a href="#cite_note-YF-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Bevegelseskonstanter">Bevegelseskonstanter</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=5" title="Rediger avsnitt: Bevegelseskonstanter" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=5" title="Rediger kildekoden til seksjonen Bevegelseskonstanter"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Et generelt system er beskrevet ved <i>N</i> <i>generelle</i> koordinater <i>q</i> = <i>(q<sub>1</sub>, q<sub>2</sub>, ... , q<sub>N</sub>)</i>. Disse behøver ikke å være komponenter av forskjellige posisjonsvektorer, men kan for eksempel oppstå ved bruk av ikke-kartesiske koordinatsystem. Begynnelsespunkt <i>A</i> og sluttpunkt <i>B</i> for bevegelsen er da begge angitt ved <i>N</i> slike koordinater. Virkningen må nå være stasjonær under variasjon <i>q<sub> n</sub>(t) → q<sub> n</sub>(t) + δq<sub> n</sub>(t)</i> av alle disse koordinatene. Dermed får man en Euler-Lagrange-ligning for hver slik dynamisk variabel, </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 {\partial L \over \partial q_{n}}-{d \over dt}{\partial L \over \partial {\dot {q}}_{n}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial L \over \partial q_{n}}-{d \over dt}{\partial L \over \partial {\dot {q}}_{n}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a580f7517ddeb3627cfb55f762057669f6fafca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.799ex; height:6.009ex;" alt="{\displaystyle {\partial L \over \partial q_{n}}-{d \over dt}{\partial L \over \partial {\dot {q}}_{n}}=0.}"></span></dd></dl> <p>Hvis disse koordinatene ikke alle er uavhengige av hverandre, men oppfyller visse bibetingelser, kan det tas hensyn til ved bruk av metoden med <a href="/wiki/Lagrange-multiplikator" title="Lagrange-multiplikator">Lagrange-multiplikator</a>. </p><p>Det siste leddet i denne ligningen inneholder den tidsderiverte av hva som kalles den <b>konjugerte</b> <a href="/wiki/Impuls_(fysikk)" class="mw-redirect" title="Impuls (fysikk)">impuls</a> <i>p<sub>n</sub></i> til koordinaten <i>q<sub>n</sub></i>, det vil si </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 p_{n}={\partial L \over \partial {\dot {q}}_{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}={\partial L \over \partial {\dot {q}}_{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/418ca85d6aaf145c97b13d0991c4f6dcfd0986be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:10.972ex; height:6.009ex;" alt="{\displaystyle p_{n}={\partial L \over \partial {\dot {q}}_{n}}.}"></span></dd></dl> <p>Euler-Lagrange-ligningen kan derfor skrives som </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 {dp_{n} \over dt}={\partial L \over \partial q_{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dp_{n} \over dt}={\partial L \over \partial q_{n}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ff949f808e7d13ef99a0fd48d2b4e9e0eb056e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.595ex; height:5.843ex;" alt="{\displaystyle {dp_{n} \over dt}={\partial L \over \partial q_{n}},}"></span></dd></dl> <p>som er en generell versjon av <a href="/wiki/Newtons_lover" class="mw-redirect" title="Newtons lover">Newtons andre lov</a>. </p><p>Hvis Lagrange-funksjonen av en eller annen grunn ikke inneholder en koordinat <i>q</i><sub> k</sub> slik at <i>∂ L/∂ q<sub>k</sub></i> = 0, så betyr det at <i>dp<sub>k</sub>/dt</i> = 0. Den tilhørende, konjugerte impuls er derfor uavhengig av tiden og dermed er </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 p_{k}=konst}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{k}=konst}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f390410cdc956b5195704a005fa1a2b835d6a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.11ex; height:2.509ex;" alt="{\displaystyle p_{k}=konst}"></span></dd></dl> <p>en <b>bevegelseskonstant</b>. Den tilsvarende variabel sies å være <b>syklisk</b>. Det kan i prinsippet være flere av dem, og de kan være til stor hjelp ved løsningen av Euler-Lagrange-ligningene for de andre koordinatene.<sup id="cite_ref-Goldstein_1-3" class="reference"><a href="#cite_note-Goldstein-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Hamilton-funksjonen">Hamilton-funksjonen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=6" title="Rediger avsnitt: Hamilton-funksjonen" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=6" title="Rediger kildekoden til seksjonen Hamilton-funksjonen"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fra den generelle <a href="/wiki/Variasjonsregning" title="Variasjonsregning">variasjonsregningen</a> vet man også at når tiden <i>t</i> ikke eksplisitt opptrer i Lagrange-funksjonen, vil størrelsen </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 H=\sum _{n}p_{n}{\dot {q}}_{n}-L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=\sum _{n}p_{n}{\dot {q}}_{n}-L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8663f8327005282221993f10defc16a2250677e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.311ex; height:5.509ex;" alt="{\displaystyle H=\sum _{n}p_{n}{\dot {q}}_{n}-L}"></span></dd></dl> <p>være en bevegelseskonstant. Generelt er </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 {dH \over dt}=\sum _{n}\left(p_{n}{\ddot {q}}_{n}+{dp_{n} \over dt}{\dot {q}}_{n}-{\partial L \over \partial q_{n}}{\dot {q}}_{n}-{\partial L \over \partial {\dot {q}}_{n}}{\ddot {q}}_{n}\right)-{\partial L \over \partial t}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>H</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dH \over dt}=\sum _{n}\left(p_{n}{\ddot {q}}_{n}+{dp_{n} \over dt}{\dot {q}}_{n}-{\partial L \over \partial q_{n}}{\dot {q}}_{n}-{\partial L \over \partial {\dot {q}}_{n}}{\ddot {q}}_{n}\right)-{\partial L \over \partial t}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5ea40cdd52d6ca1f0b9a5d56bf68d0a2ecffe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.492ex; height:6.676ex;" alt="{\displaystyle {dH \over dt}=\sum _{n}\left(p_{n}{\ddot {q}}_{n}+{dp_{n} \over dt}{\dot {q}}_{n}-{\partial L \over \partial q_{n}}{\dot {q}}_{n}-{\partial L \over \partial {\dot {q}}_{n}}{\ddot {q}}_{n}\right)-{\partial L \over \partial t}.}"></span></dd></dl> <p>Fra Euler-Lagrange-ligningen ser man her at de to midtre leddene i parentesen kansellerer hverandre. Det gjør også første og siste ledd i parentesen ut fra definisjonen av konjugert impuls. Dermed er </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 {dH \over dt}=-{\partial L \over \partial t}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>H</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dH \over dt}=-{\partial L \over \partial t}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e832b71434176c7dd7703feae853229260d447f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.406ex; height:5.509ex;" alt="{\displaystyle {dH \over dt}=-{\partial L \over \partial t}.}"></span></dd></dl> <p>Derfor er <i>H</i> = <i>E</i> en konstant når <i>∂ L/∂ t</i> = 0. Denne bevegelseskonstanten er ikke noe annet enn den totale <a href="/wiki/Energi" title="Energi">energien</a> til systemet. Tid og energi er konjugerte variable. </p><p>I eksempelet over med den harmoniske oscillator finner vi den konjugerte impulsen <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 p=m{\dot {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=m{\dot {q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329a0d9adb0a5605a998e2f77be8bcfcdfacd1c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.775ex; height:2.509ex;" alt="{\displaystyle p=m{\dot {q}}}"></span> slik at </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=m{\dot {q}}^{2}-{1 \over 2}m{\dot {q}}^{2}+{1 \over 2}kq^{2}={1 \over 2}m{\dot {q}}^{2}+{1 \over 2}kq^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=m{\dot {q}}^{2}-{1 \over 2}m{\dot {q}}^{2}+{1 \over 2}kq^{2}={1 \over 2}m{\dot {q}}^{2}+{1 \over 2}kq^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d06ddf8689918359300caf67df7137873f3dcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.24ex; height:5.176ex;" alt="{\displaystyle E=m{\dot {q}}^{2}-{1 \over 2}m{\dot {q}}^{2}+{1 \over 2}kq^{2}={1 \over 2}m{\dot {q}}^{2}+{1 \over 2}kq^{2},}"></span></dd></dl> <p>som er summen <i>E = T + V</i> av kinetisk og potensiell energi. Uttrykt ved impulsen er </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 H(q,p)={p^{2} \over 2m}+{1 \over 2}kq^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(q,p)={p^{2} \over 2m}+{1 \over 2}kq^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3cf81af686ae554d648344037b0777458e2fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.467ex; height:5.676ex;" alt="{\displaystyle H(q,p)={p^{2} \over 2m}+{1 \over 2}kq^{2}}"></span></dd></dl> <p>den totale energien til oscillatoren. Dette er et eksempel på det som generelt kalles en <b>Hamilton-funksjon</b>. Den danner grunnlaget for <a href="/wiki/Hamilton-mekanikk" title="Hamilton-mekanikk">Hamilton-mekanikken</a> som gir en alternativ beskrivelse av mekaniske system. </p> <div class="mw-heading mw-heading2"><h2 id="Sentralsymmetrisk_potensial">Sentralsymmetrisk potensial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=7" title="Rediger avsnitt: Sentralsymmetrisk potensial" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=7" title="Rediger kildekoden til seksjonen Sentralsymmetrisk potensial"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I mange realistiske tilfeller vil en partikkel bevege seg med en potensiell energi <i>V</i> som ikke avhenger av retningen til dens posisjonsvektor <b>r</b>, men bare av dens lengde <i>r</i> = |<b>r</b>|. Da er <i>V = V(r)</i> og systemet er symmetrisk under rotasjoner rundt origo. Kraften <b>F</b> = - <b>∇</b><i>V</i> peker da i radiell retning slik at <a href="/wiki/Dreiemoment" title="Dreiemoment">dreiemomentet</a> <b>r</b> × <b>F</b> på den er null. Dermed er dens <a href="/wiki/Dreieimpuls" class="mw-redirect" title="Dreieimpuls">dreieimpuls</a> <b>L</b>= <b>r</b> × <b>p</b> en konstant vektor. </p><p>Man kan nå velge et aksekors med <i>z</i>-aksen langs dreieimpulsen. Da vil bevegelsen foregå i <i>(x,y)</i> - planet og kan beskrives ved bruk av <a href="/wiki/Polarkoordinatsystem" title="Polarkoordinatsystem">polarkoordinater</a> (<i>r,φ</i>) i dette planet. Da er <i>x</i> = <i>r cosφ</i> og <i>y</i> = <i>r sinφ</i>. Ved direkte derivasjon finner man hastighetskomponenten <i>dx/dt</i> = <i>(dr/dt) cosφ</i> - <i>r sinφ(dφ/dt</i>) og tilsvarende for komponenten <i>dy/dt</i>. Kvadrerer man disse uttrykkene og adderer sammen, finner man Lagrange-funksjonen </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 L={m \over 2}{\Big (}{\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}{\Big )}-V(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={m \over 2}{\Big (}{\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}{\Big )}-V(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f46c10755eb44a1e4c34305ae8f9e9c761d2ea2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.63ex; height:4.843ex;" alt="{\displaystyle L={m \over 2}{\Big (}{\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}{\Big )}-V(r)}"></span></dd></dl> <p>i dette koordinatsystemet. Vinkelen <i>φ</i> opptrer ikke her og er dermed en syklisk variabel. Den konjugerte impulsen </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 p_{\phi }={\partial L \over \partial {\dot {\phi }}}=mr^{2}{\dot {\phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\phi }={\partial L \over \partial {\dot {\phi }}}=mr^{2}{\dot {\phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2289e4004288e4322c20e2263a6abf50cea917bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:18.015ex; height:6.509ex;" alt="{\displaystyle p_{\phi }={\partial L \over \partial {\dot {\phi }}}=mr^{2}{\dot {\phi }}}"></span></dd></dl> <p>som er dreieimpulsen, er derfor en bevegelseskonstant <i>p<sub>φ</sub></i> = <i>k</i>. I forbindelse med planeters bevegelse, er </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 {dA \over dt}={1 \over 2}r^{2}{\dot {\phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>A</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dA \over dt}={1 \over 2}r^{2}{\dot {\phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3657aa9284ed3a5f65153e333395de99a93c76d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.462ex; height:5.509ex;" alt="{\displaystyle {dA \over dt}={1 \over 2}r^{2}{\dot {\phi }}}"></span></dd></dl> <p>«flatehastigheten» som banen beskriver. At dreieimpulsen er konstant, betyr altså at denne flatehastigheten er uforanderlig. Dette er <a href="/wiki/Keplers_lover" title="Keplers lover">Keplers andre lov</a>.<sup id="cite_ref-YF_3-1" class="reference"><a href="#cite_note-YF-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Resten av dynamikken til systemet er beskrevet ved den radielle koordinaten. Impulsen konjugert til <i>r</i> er </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 p_{r}={\partial L \over \partial {\dot {r}}}=m{\dot {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{r}={\partial L \over \partial {\dot {r}}}=m{\dot {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1c306f1c852d71493c52dd057b2c6ce69c22a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:15.499ex; height:5.509ex;" alt="{\displaystyle p_{r}={\partial L \over \partial {\dot {r}}}=m{\dot {r}}}"></span></dd></dl> <p>mens </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 {\partial L \over \partial r}=mr{\dot {\phi }}^{2}-{dV \over dr},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>V</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial L \over \partial r}=mr{\dot {\phi }}^{2}-{dV \over dr},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9213fb5ba5033178df44eb6c0327d5f269da2bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.773ex; height:5.509ex;" alt="{\displaystyle {\partial L \over \partial r}=mr{\dot {\phi }}^{2}-{dV \over dr},}"></span></dd></dl> <p>slik at Euler-Lagrange-ligningen blir </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 m{\ddot {r}}={k^{2} \over mr^{3}}-{dV \over dr}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>V</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\ddot {r}}={k^{2} \over mr^{3}}-{dV \over dr}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b302cb4ea3700c1ccb5ea01e029aa82b6d8fb8d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.736ex; height:6.009ex;" alt="{\displaystyle m{\ddot {r}}={k^{2} \over mr^{3}}-{dV \over dr}.}"></span></dd></dl> <p>Dette er en andre ordens <a href="/wiki/Differensialligning" title="Differensialligning">differensialligning</a> som kan løses analytisk for Newtons gravitasjonslov <i>V = - GmM/r</i> hvor <i>M</i> er sentralmassen og <i>G</i> er <a href="/wiki/Gravitasjonskonstanten" class="mw-redirect" title="Gravitasjonskonstanten">gravitasjonskonstanten</a>. Første ledd på høyre side er det effektive potensialet som skyldes <a href="/wiki/Sentrifugalkraft" title="Sentrifugalkraft">sentrifugalkraften</a>. De bundne løsningene gir <a href="/wiki/Ellipse" title="Ellipse">ellipsebaner</a> som er innholdet av <a href="/wiki/Keplers_lover" title="Keplers lover">Keplers første lov</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Geodetisk_bevegelse">Geodetisk bevegelse</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=8" title="Rediger avsnitt: Geodetisk bevegelse" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=8" title="Rediger kildekoden til seksjonen Geodetisk bevegelse"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Generelt ved bruk av <a href="/wiki/Krumlinjete_koordinater" title="Krumlinjete koordinater">krumlinjete koordinater</a> <i>x<sup>μ</sup></i>  skrives den <a href="/wiki/Kinetisk_energi" title="Kinetisk energi">kinetiske energien</a> for en ikke-relativistisk partikkel som </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 T={1 \over 2}mv^{2}={1 \over 2}m{\Big (}{ds \over dt}{\Big )}^{2}={1 \over 2}mg_{\mu \nu }{\dot {x}}^{\mu }{\dot {x}}^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>s</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={1 \over 2}mv^{2}={1 \over 2}m{\Big (}{ds \over dt}{\Big )}^{2}={1 \over 2}mg_{\mu \nu }{\dot {x}}^{\mu }{\dot {x}}^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4594828a3300c4b2cf58bba0adc53bb62d077d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.393ex; height:5.509ex;" alt="{\displaystyle T={1 \over 2}mv^{2}={1 \over 2}m{\Big (}{ds \over dt}{\Big )}^{2}={1 \over 2}mg_{\mu \nu }{\dot {x}}^{\mu }{\dot {x}}^{\nu }}"></span></dd></dl> <p>når man gjør bruk av <a href="/wiki/Einsteins_summekonvensjon" title="Einsteins summekonvensjon">Einsteins summekonvensjon</a> og summerer over par med like indekser. Her er <i>g<sub>μν</sub></i>  den <a href="/wiki/Metrisk_tensor" title="Metrisk tensor">metriske tensoren</a> for disse koordinatene og <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^{\mu }={\dot {x}}^{\mu }=dx^{\mu }/dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{\mu }={\dot {x}}^{\mu }=dx^{\mu }/dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72ef01066cfd0b9d09d5ebf0443d79068b72862d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.088ex; height:2.843ex;" alt="{\displaystyle v^{\mu }={\dot {x}}^{\mu }=dx^{\mu }/dt}"></span> er komponentene til hastigheten. Hvis partikkelen har en <a href="/wiki/Potensiell_energi" title="Potensiell energi">potensiell energi</a> som er uavhengig av hastigheten, er Lagrange-funksjonen </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 L={1 \over 2}mg_{\mu \nu }(x)v^{\mu }v^{\nu }-V(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={1 \over 2}mg_{\mu \nu }(x)v^{\mu }v^{\nu }-V(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0186591632d8973f35685b9854112f095dfd2ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.412ex; height:5.176ex;" alt="{\displaystyle L={1 \over 2}mg_{\mu \nu }(x)v^{\mu }v^{\nu }-V(x)}"></span></dd></dl> <p>Herav finnes den konjugerte impulsen <span class="nowrap"><i>p<sub>λ</sub> = ∂ L/∂ v<sup>λ</sup> = mg<sub>λμ</sub>v<sup>μ</sup></i></span>. Euler-Lagrange-ligningen tar da formen <i>dp<sub>λ</sub>/dτ = ∂ L/∂ x<sup>λ</sup></i> hvor </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 {dp_{\lambda } \over dt}=mg_{\lambda \mu }{dv^{\mu } \over dt}+m{\partial g_{\lambda \mu } \over \partial x^{\nu }}v^{\mu }v^{\nu }=mg_{\lambda \mu }{dv^{\mu } \over dt}+{m \over 2}{\Big (}{\partial g_{\lambda \mu } \over \partial x^{\nu }}+{\partial g_{\lambda \nu } \over \partial x^{\mu }}{\Big )}v^{\mu }v^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>μ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>m</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>μ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dp_{\lambda } \over dt}=mg_{\lambda \mu }{dv^{\mu } \over dt}+m{\partial g_{\lambda \mu } \over \partial x^{\nu }}v^{\mu }v^{\nu }=mg_{\lambda \mu }{dv^{\mu } \over dt}+{m \over 2}{\Big (}{\partial g_{\lambda \mu } \over \partial x^{\nu }}+{\partial g_{\lambda \nu } \over \partial x^{\mu }}{\Big )}v^{\mu }v^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155699831a0f78c4901db7c1a8cf6a3ba24cb9ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:71.67ex; height:5.843ex;" alt="{\displaystyle {dp_{\lambda } \over dt}=mg_{\lambda \mu }{dv^{\mu } \over dt}+m{\partial g_{\lambda \mu } \over \partial x^{\nu }}v^{\mu }v^{\nu }=mg_{\lambda \mu }{dv^{\mu } \over dt}+{m \over 2}{\Big (}{\partial g_{\lambda \mu } \over \partial x^{\nu }}+{\partial g_{\lambda \nu } \over \partial x^{\mu }}{\Big )}v^{\mu }v^{\nu }}"></span></dd></dl> <p>etter å ha skrevet det siste leddet på en mer symmetrisk form. I tillegg er </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 {\partial L \over \partial x^{\lambda }}={m \over 2}{\partial g_{\mu \nu } \over \partial x^{\lambda }}v^{\mu }v^{\nu }-{\partial V \over \partial x^{\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial L \over \partial x^{\lambda }}={m \over 2}{\partial g_{\mu \nu } \over \partial x^{\lambda }}v^{\mu }v^{\nu }-{\partial V \over \partial x^{\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab24074bd178b5ad6a4268177b77ea3065d2d81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.104ex; height:6.009ex;" alt="{\displaystyle {\partial L \over \partial x^{\lambda }}={m \over 2}{\partial g_{\mu \nu } \over \partial x^{\lambda }}v^{\mu }v^{\nu }-{\partial V \over \partial x^{\lambda }}}"></span></dd></dl> <p>slik at den generelle bevegelsesligningen blir </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 m\left(g_{\lambda \mu }{dv^{\mu } \over dt}+\Gamma _{\lambda \mu \nu }v^{\mu }v^{\nu }\right)=-{\partial V \over \partial x^{\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\left(g_{\lambda \mu }{dv^{\mu } \over dt}+\Gamma _{\lambda \mu \nu }v^{\mu }v^{\nu }\right)=-{\partial V \over \partial x^{\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8002dc0be12eab719eb90b40ff20079d8f768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.052ex; height:6.176ex;" alt="{\displaystyle m\left(g_{\lambda \mu }{dv^{\mu } \over dt}+\Gamma _{\lambda \mu \nu }v^{\mu }v^{\nu }\right)=-{\partial V \over \partial x^{\lambda }}}"></span></dd></dl> <p>Den kan betraktes som <a href="/wiki/Newtons_bevegelseslover" title="Newtons bevegelseslover">Newtons andre ligning</a> skrevet i et vilkårlig koordinatsystem. Innholdet i parentesen er det tilfellet et uttrykk for <a href="/wiki/Akselerasjon" title="Akselerasjon">akselerasjonen</a> hvor de deriverte av den metriske tensoren inngår i kombinasjonen </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 \Gamma _{\lambda \mu \nu }={1 \over 2}{\Big (}{\partial g_{\lambda \mu } \over \partial x^{\nu }}+{\partial g_{\lambda \nu } \over \partial x^{\mu }}-{\partial g_{\mu \nu } \over \partial x^{\lambda }}{\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>μ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{\lambda \mu \nu }={1 \over 2}{\Big (}{\partial g_{\lambda \mu } \over \partial x^{\nu }}+{\partial g_{\lambda \nu } \over \partial x^{\mu }}-{\partial g_{\mu \nu } \over \partial x^{\lambda }}{\Big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1155fadbbe26df09152ff8986594c72189d2b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:34.187ex; height:6.009ex;" alt="{\displaystyle \Gamma _{\lambda \mu \nu }={1 \over 2}{\Big (}{\partial g_{\lambda \mu } \over \partial x^{\nu }}+{\partial g_{\lambda \nu } \over \partial x^{\mu }}-{\partial g_{\mu \nu } \over \partial x^{\lambda }}{\Big )}}"></span></dd></dl> <p>som er <a href="/wiki/Tensor#Levi-Civita-konneksjonen" title="Tensor">Christoffel-symbolet</a> av første type. Det er symmetrisk i de to siste indeksene. </p><p>For en fri partikkel er potensialet <i>V</i> = 0. Dens bevegelse er derfor gitt ved en enklere differensialligning som finnes ved å multiplisere med den kontravariante metrikken <i>g<sup>σλ</sup></i> på begge sider. Det gir </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 {dv^{\mu } \over dt}+\Gamma _{\;\mu \nu }^{\lambda }v^{\mu }v^{\nu }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thickmathspace" /> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msubsup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dv^{\mu } \over dt}+\Gamma _{\;\mu \nu }^{\lambda }v^{\mu }v^{\nu }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58092dda9b23502520073991d0e60aebba534545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.279ex; height:5.509ex;" alt="{\displaystyle {dv^{\mu } \over dt}+\Gamma _{\;\mu \nu }^{\lambda }v^{\mu }v^{\nu }=0}"></span></dd></dl> <p>etter å ha innført det andre Christoffel-symbolet <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 \Gamma _{\;\mu \nu }^{\sigma }=g^{\sigma \lambda }\Gamma _{\lambda \mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thickmathspace" /> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mi>λ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{\;\mu \nu }^{\sigma }=g^{\sigma \lambda }\Gamma _{\lambda \mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43d99fbf270e8e33af35ec274ef330d25465e4f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.046ex; height:3.343ex;" alt="{\displaystyle \Gamma _{\;\mu \nu }^{\sigma }=g^{\sigma \lambda }\Gamma _{\lambda \mu \nu }}"></span>. Dette resultatet kalles den <a href="/wiki/Geodetisk_kurve" title="Geodetisk kurve">geodetiske ligningen</a> og beskriver den korteste veien som forbinder begynnelsespunkt og sluttpunkt. En slik spesiell bane kalles en <i>geodetisk kurve</i> og karakteriserer bevegelsen til en fri partikkel. Dette gjelder hvis den for eksempel beveger seg på en <a href="/wiki/Differensiell_flategeometri" title="Differensiell flategeometri">krum flate</a> eller i <a href="/wiki/Kovariant_relativitetsteori#Relativistisk_partikkel_i_gravitasjonsfelt" title="Kovariant relativitetsteori">generell relativitetsteori</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Eksempel">Eksempel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=9" title="Rediger avsnitt: Eksempel" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=9" title="Rediger kildekoden til seksjonen Eksempel"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Betrakter man en fri partikkel som beveger seg i et plan og beskrevet ved polarkoordinatene (<i>r,φ</i>), er Lagrange-funksjonen </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 L={m \over 2}\left({\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={m \over 2}\left({\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39e8ce2d565f14a6af4fa5776628a3d184302d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.531ex; height:4.843ex;" alt="{\displaystyle L={m \over 2}\left({\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}\right)}"></span></dd></dl> <p>Euler-Lagrange-ligningen for den sykliske variable <i>φ </i> kan skrives som </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 mr^{2}{d\phi \over dt}=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mr^{2}{d\phi \over dt}=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3cf6a4691a698caaeff45811cde7d2fd0b9e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.89ex; height:5.509ex;" alt="{\displaystyle mr^{2}{d\phi \over dt}=k}"></span></dd></dl> <p>Formen til banen følger fra ligningen </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 m{d^{2}r \over dt^{2}}={k^{2} \over mr^{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{d^{2}r \over dt^{2}}={k^{2} \over mr^{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f54ec9276d6199b1a974b975ded8ec1996680e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.275ex; height:6.009ex;" alt="{\displaystyle m{d^{2}r \over dt^{2}}={k^{2} \over mr^{3}}}"></span></dd></dl> <p>for den radielle variable. Ved å benytte at <i>d/dt</i> = (<i>dφ</i>/<i>dt</i>)<i>d/dφ</i>, kan den nå omformes til </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 {d \over d\phi }\left({1 \over r^{2}}{dr \over d\phi }\right)={1 \over r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over d\phi }\left({1 \over r^{2}}{dr \over d\phi }\right)={1 \over r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/211a18fc755fe77c1c465e647500523381bdfdd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.719ex; height:6.176ex;" alt="{\displaystyle {d \over d\phi }\left({1 \over r^{2}}{dr \over d\phi }\right)={1 \over r}}"></span></dd></dl> <p>som er den geodetiske ligningen for banen. Ved å innføre <i>u</i> = 1/<i>r</i>  som ny banevariabel, tar den formen </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 {d^{2}u \over d\phi ^{2}}+u=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d^{2}u \over d\phi ^{2}}+u=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8ffae96d8722dc6e86f0476d7d47df9b202cd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.923ex; height:6.343ex;" alt="{\displaystyle {d^{2}u \over d\phi ^{2}}+u=0}"></span></dd></dl> <p>som er den <a href="/wiki/Harmonisk_oscillator" title="Harmonisk oscillator">harmoniske svingeligningen</a>. Den generelle løsningen kan skrives som <i>u</i> = (1/<i>b</i>)  cos(<i>φ - φ</i><sub>0</sub>)  hvor <i>b</i> og <i>φ</i><sub>0</sub>  er integrasjonskonstanter. Bevegelsen til den frie partikkelen er derfor gitt ved baneligningen </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 r\cos(\phi -\phi _{0})=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>−</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\cos(\phi -\phi _{0})=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf2ece6ac6580142cc57a7349f00565fefef0fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.118ex; height:2.843ex;" alt="{\displaystyle r\cos(\phi -\phi _{0})=b}"></span></dd></dl> <p>som beskriver en rett linje i polarkoordinater. Det punkt på banen som er nærmest til origo, har avstand <i>b</i> og ligger i retning <i>φ</i><sub>0</sub>. Geometrisk følger dette resultatet fra cosinus til vinkelen <span class="nowrap"><i>θ</i> - <i>θ</i><sub>0</sub> </span> i den rettvinklet trekanten som har hosliggende <a href="/wiki/Katet" title="Katet">katet</a> <i>b </i> og <a href="/wiki/Hypotenus" title="Hypotenus">hypotenus</a> <i>r</i>. </p><p>Den geodetiske ligningen er vanligvis en differensialligning av andre orden. Men når partikkelen har en konstant energi som her, kan man i stedet finne en geodetisk ligning av første orden mer direkte fra den konstante energien <i>E</i>. For en fri partikkel er den lik med Lagrange-funksjonen <i>L</i>. Da er </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 \left({k \over mr^{2}}{dr \over d\phi }\right)^{2}+r^{2}\left({k \over mr^{2}}\right)^{2}={2E \over m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>E</mi> </mrow> <mi>m</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({k \over mr^{2}}{dr \over d\phi }\right)^{2}+r^{2}\left({k \over mr^{2}}\right)^{2}={2E \over m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff63d4a97894ce34310b9ee5336692abad8b3b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.163ex; height:6.509ex;" alt="{\displaystyle \left({k \over mr^{2}}{dr \over d\phi }\right)^{2}+r^{2}\left({k \over mr^{2}}\right)^{2}={2E \over m}}"></span></dd></dl> <p>Det betyr at </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 {d\phi \over dr}={1 \over r{\sqrt {Br^{2}-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>B</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d\phi \over dr}={1 \over r{\sqrt {Br^{2}-1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0f92a93a1ad9ab516c5790ba14155ae75dbd9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.614ex; height:6.676ex;" alt="{\displaystyle {d\phi \over dr}={1 \over r{\sqrt {Br^{2}-1}}}}"></span></dd></dl> <p>etter å ha innført konstanten <i>B</i> = 2<i>mE</i>/<i>k</i><sup>2</sup> = 1/<i>b</i><sup>2</sup>. Skriver man nå <i>r</i> = <i>b</i>/<i>x</i>, forenkles denne ligningen til </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 {d\phi \over dx}=-{1 \over {\sqrt {1-x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d\phi \over dx}=-{1 \over {\sqrt {1-x^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a8cdc4f9f4b6670893b4c44668a6de27d52c6bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.891ex; height:6.676ex;" alt="{\displaystyle {d\phi \over dx}=-{1 \over {\sqrt {1-x^{2}}}}}"></span></dd></dl> <p>som kan løses ved direkte integrasjon. Det gir <i>φ</i> = arccos<i>x</i> + <i>φ</i><sub>0</sub>  eller <i>x</i> = cos(<i>φ - φ</i><sub>0</sub>). Dette er igjen samme ligning for den rette linjen som partikkelen følger.<sup id="cite_ref-Goldstine_2-1" class="reference"><a href="#cite_note-Goldstine-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Se_også"><span id="Se_ogs.C3.A5"></span>Se også</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=10" title="Rediger avsnitt: Se også" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=10" title="Rediger kildekoden til seksjonen Se også"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Variasjonsregning" title="Variasjonsregning">Variasjonsregning</a></li> <li><a href="/wiki/Hamilton-mekanikk" title="Hamilton-mekanikk">Hamilton-mekanikk</a></li> <li><a href="/wiki/Virkningsprinsipp" title="Virkningsprinsipp">Virkningsprinsipp</a></li> <li><a href="/wiki/Hamiltons_virkningsprinsipp" title="Hamiltons virkningsprinsipp">Hamiltons virkningsprinsipp</a></li> <li><a href="/wiki/Maupertuis%27_virkningsprinsipp" class="mw-redirect" title="Maupertuis' virkningsprinsipp">Maupertuis' virkningsprinsipp</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referanser">Referanser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=11" title="Rediger avsnitt: Referanser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=11" title="Rediger kildekoden til seksjonen Referanser"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Goldstein-1"><b>^</b> <a href="#cite_ref-Goldstein_1-0"><sup>a</sup></a> <a href="#cite_ref-Goldstein_1-1"><sup>b</sup></a> <a href="#cite_ref-Goldstein_1-2"><sup>c</sup></a> <a href="#cite_ref-Goldstein_1-3"><sup>d</sup></a> <span class="reference-text"> H. Goldstein, <i>Classical Mechanics</i>, Addidon-Wesley Publishing Company, Reading, Massachusetts (1959).</span> </li> <li id="cite_note-Goldstine-2"><b>^</b> <a href="#cite_ref-Goldstine_2-0"><sup>a</sup></a> <a href="#cite_ref-Goldstine_2-1"><sup>b</sup></a> <span class="reference-text"> H. Goldstine: <i>A History of the Calculus of Variations from the 17th through the 19th Century</i>, Springer, New York (1980). <a href="/wiki/Spesial:Bokkilder/1461381068" class="internal mw-magiclink-isbn">ISBN 1-4613-8106-8</a>.</span> </li> <li id="cite_note-YF-3"><b>^</b> <a href="#cite_ref-YF_3-0"><sup>a</sup></a> <a href="#cite_ref-YF_3-1"><sup>b</sup></a> <span class="reference-text"> H.D. Young and R.A. Freedman, <i>University Physics</i>, Addison-Wesley, New York (2008). <a href="/wiki/Spesial:Bokkilder/9780321501301" class="internal mw-magiclink-isbn">ISBN 978-0-321-50130-1</a>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Litteratur">Litteratur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange-mekanikk&veaction=edit&section=12" title="Rediger avsnitt: Litteratur" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lagrange-mekanikk&action=edit&section=12" title="Rediger kildekoden til seksjonen Litteratur"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A. Sommerfeld, <i>Vorlesungen über Theoretische Physik, Band I: Mechanik</i>, Akademische Verlagsgellschaft, Leipzig (1964).</li> <li>L. N. Hand and J. D. 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