Hamiltona mekaniko

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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Krampoj de Poisson</span> </div> </a> <ul id="toc-Krampoj_de_Poisson-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Principo_de_senmova_ago_por_hamiltonaj_sistemoj" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Principo_de_senmova_ago_por_hamiltonaj_sistemoj"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Principo de senmova ago por hamiltonaj sistemoj</span> </div> </a> <ul id="toc-Principo_de_senmova_ago_por_hamiltonaj_sistemoj-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teoremo_de_Liouville" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Teoremo_de_Liouville"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Teoremo de Liouville</span> </div> </a> <ul id="toc-Teoremo_de_Liouville-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notoj" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notoj"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notoj</span> </div> </a> <ul id="toc-Notoj-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencoj" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referencoj"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Referencoj</span> </div> </a> <button aria-controls="toc-Referencoj-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>Baskuli Referencoj subsekcio</span> </button> <ul id="toc-Referencoj-sublist" class="vector-toc-list"> <li id="toc-Bibliografio" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliografio"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Bibliografio</span> </div> </a> <ul id="toc-Bibliografio-sublist" class="vector-toc-list"> </ul> </li> </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="Enhavo" 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="Ŝaltu la enhavtabelon" > <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">Ŝaltu la enhavtabelon</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">Hamiltona mekaniko</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="Iru al artikolo en alia lingvo. Havebla en 38 lingvo" > <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-38" 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">38 lingvoj</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Hamiltoniese_meganika" title="Hamiltoniese meganika — afrikansa" lang="af" hreflang="af" data-title="Hamiltoniese meganika" data-language-autonym="Afrikaans" data-language-local-name="afrikansa" 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%87%D8%A7%D9%85%D9%84%D8%AA%D9%88%D9%86%D9%8A" title="ميكانيكا هاملتوني — araba" lang="ar" hreflang="ar" data-title="ميكانيكا هاملتوني" data-language-autonym="العربية" data-language-local-name="araba" 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%93%D0%B0%D0%BC%D1%96%D0%BB%D1%8C%D1%82%D0%B0%D0%BD%D0%B0%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Гамільтанава механіка — belorusa" lang="be" hreflang="be" data-title="Гамільтанава механіка" data-language-autonym="Беларуская" data-language-local-name="belorusa" 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%A5%D0%B0%D0%BC%D0%B8%D0%BB%D1%82%D0%BE%D0%BD%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Хамилтонова механика — bulgara" lang="bg" hreflang="bg" data-title="Хамилтонова механика" data-language-autonym="Български" data-language-local-name="bulgara" 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_hamiltoniana" title="Formulació hamiltoniana — kataluna" lang="ca" hreflang="ca" data-title="Formulació hamiltoniana" data-language-autonym="Català" data-language-local-name="kataluna" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hamiltonovsk%C3%A1_formulace_mechaniky" title="Hamiltonovská formulace mechaniky — ĉeĥa" lang="cs" hreflang="cs" data-title="Hamiltonovská formulace mechaniky" data-language-autonym="Čeština" data-language-local-name="ĉeĥa" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B0%D0%BC%D0%B8%D0%BB%D1%8C%D1%82%D0%BE%D0%BD%D0%BB%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Гамильтонла механика — ĉuvaŝa" lang="cv" hreflang="cv" data-title="Гамильтонла механика" data-language-autonym="Чӑвашла" data-language-local-name="ĉuvaŝa" 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/Hamiltonsche_Mechanik" title="Hamiltonsche Mechanik — germana" lang="de" hreflang="de" data-title="Hamiltonsche Mechanik" data-language-autonym="Deutsch" data-language-local-name="germana" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics — angla" lang="en" hreflang="en" data-title="Hamiltonian mechanics" data-language-autonym="English" data-language-local-name="angla" 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_hamiltoniana" title="Mecánica hamiltoniana — hispana" lang="es" hreflang="es" data-title="Mecánica hamiltoniana" data-language-autonym="Español" data-language-local-name="hispana" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Hamiltoni_mehaanika" title="Hamiltoni mehaanika — estona" lang="et" hreflang="et" data-title="Hamiltoni mehaanika" data-language-autonym="Eesti" data-language-local-name="estona" class="interlanguage-link-target"><span>Eesti</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%87%D9%85%DB%8C%D9%84%D8%AA%D9%88%D9%86%DB%8C" title="مکانیک همیلتونی — persa" lang="fa" hreflang="fa" data-title="مکانیک همیلتونی" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hamiltonin_mekaniikka" title="Hamiltonin mekaniikka — finna" lang="fi" hreflang="fi" data-title="Hamiltonin mekaniikka" data-language-autonym="Suomi" data-language-local-name="finna" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9canique_hamiltonienne" title="Mécanique hamiltonienne — franca" lang="fr" hreflang="fr" data-title="Mécanique hamiltonienne" data-language-autonym="Français" data-language-local-name="franca" 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_hamiltoniana" title="Mecánica hamiltoniana — galega" lang="gl" hreflang="gl" data-title="Mecánica hamiltoniana" data-language-autonym="Galego" data-language-local-name="galega" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B9%E0%A5%88%E0%A4%AE%E0%A4%BF%E0%A4%B2%E0%A5%8D%E0%A4%9F%E0%A4%A8%E0%A5%80_%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="हैमिल्टनी यांत्रिकी — hinda" lang="hi" hreflang="hi" data-title="हैमिल्टनी यांत्रिकी" data-language-autonym="हिन्दी" data-language-local-name="hinda" 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_Hamiltonian" title="Mekanika Hamiltonian — indonezia" lang="id" hreflang="id" data-title="Mekanika Hamiltonian" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezia" 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_hamiltoniana" title="Meccanica hamiltoniana — itala" lang="it" hreflang="it" data-title="Meccanica hamiltoniana" data-language-autonym="Italiano" data-language-local-name="itala" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%8F%E3%83%9F%E3%83%AB%E3%83%88%E3%83%B3%E5%8A%9B%E5%AD%A6" title="ハミルトン力学 — japana" lang="ja" hreflang="ja" data-title="ハミルトン力学" data-language-autonym="日本語" data-language-local-name="japana" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%B4%EB%B0%80%ED%84%B4_%EC%97%AD%ED%95%99" title="해밀턴 역학 — korea" lang="ko" hreflang="ko" data-title="해밀턴 역학" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B9%E0%B4%BE%E0%B4%AE%E0%B4%BF%E0%B5%BD%E0%B4%9F%E0%B5%8D%E0%B4%9F%E0%B5%8B%E0%B4%A3%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="ഹാമിൽട്ടോണിയൻ ബലതന്ത്രം — malajalama" lang="ml" hreflang="ml" data-title="ഹാമിൽട്ടോണിയൻ ബലതന്ത്രം" data-language-autonym="മലയാളം" data-language-local-name="malajalama" 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/Hamiltonformalisme" title="Hamiltonformalisme — nederlanda" lang="nl" hreflang="nl" data-title="Hamiltonformalisme" data-language-autonym="Nederlands" data-language-local-name="nederlanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Hamilton-mekanikk" title="Hamilton-mekanikk — dannorvega" lang="nb" hreflang="nb" data-title="Hamilton-mekanikk" data-language-autonym="Norsk bokmål" data-language-local-name="dannorvega" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B9%E0%A9%88%E0%A8%AE%E0%A8%BF%E0%A8%B2%E0%A8%9F%E0%A9%8B%E0%A8%A8%E0%A9%80%E0%A8%85%E0%A8%A8_%E0%A8%AE%E0%A8%95%E0%A9%88%E0%A8%A8%E0%A8%BF%E0%A8%95%E0%A8%B8" title="ਹੈਮਿਲਟੋਨੀਅਨ ਮਕੈਨਿਕਸ — panĝaba" lang="pa" hreflang="pa" data-title="ਹੈਮਿਲਟੋਨੀਅਨ ਮਕੈਨਿਕਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panĝaba" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Mechanika_Hamiltona" title="Mechanika Hamiltona — pola" lang="pl" hreflang="pl" data-title="Mechanika Hamiltona" data-language-autonym="Polski" data-language-local-name="pola" 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_hamiltoniana" title="Mecânica hamiltoniana — portugala" lang="pt" hreflang="pt" data-title="Mecânica hamiltoniana" data-language-autonym="Português" data-language-local-name="portugala" 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_hamiltonian%C4%83" title="Mecanică hamiltoniană — rumana" lang="ro" hreflang="ro" data-title="Mecanică hamiltoniană" data-language-autonym="Română" data-language-local-name="rumana" 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%93%D0%B0%D0%BC%D0%B8%D0%BB%D1%8C%D1%82%D0%BE%D0%BD%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Гамильтонова механика — rusa" lang="ru" hreflang="ru" data-title="Гамильтонова механика" data-language-autonym="Русский" data-language-local-name="rusa" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics — Simple English" lang="en-simple" hreflang="en-simple" data-title="Hamiltonian mechanics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Hamiltonova_mehanika" title="Hamiltonova mehanika — slovena" lang="sl" hreflang="sl" data-title="Hamiltonova mehanika" data-language-autonym="Slovenščina" data-language-local-name="slovena" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Mekanika_e_Hamiltonit" title="Mekanika e Hamiltonit — albana" lang="sq" hreflang="sq" data-title="Mekanika e Hamiltonit" data-language-autonym="Shqip" data-language-local-name="albana" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hamiltonsk_mekanik" title="Hamiltonsk mekanik — sveda" lang="sv" hreflang="sv" data-title="Hamiltonsk mekanik" data-language-autonym="Svenska" data-language-local-name="sveda" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B9%81%E0%B8%AE%E0%B8%A1%E0%B8%B4%E0%B8%A5%E0%B8%95%E0%B8%B1%E0%B8%99" title="กลศาสตร์แฮมิลตัน — taja" lang="th" hreflang="th" data-title="กลศาสตร์แฮมิลตัน" data-language-autonym="ไทย" data-language-local-name="taja" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hamilton_mekani%C4%9Fi" title="Hamilton mekaniği — turka" lang="tr" hreflang="tr" data-title="Hamilton mekaniği" data-language-autonym="Türkçe" data-language-local-name="turka" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0_%D0%93%D0%B0%D0%BC%D1%96%D0%BB%D1%8C%D1%82%D0%BE%D0%BD%D0%B0" title="Механіка Гамільтона — ukraina" lang="uk" hreflang="uk" data-title="Механіка Гамільтона" data-language-autonym="Українська" data-language-local-name="ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Gamilton_mexanikasi" title="Gamilton mexanikasi — uzbeka" lang="uz" hreflang="uz" data-title="Gamilton mexanikasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeka" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_Hamilton" title="Cơ học Hamilton — vjetnama" lang="vi" hreflang="vi" data-title="Cơ học Hamilton" data-language-autonym="Tiếng Việt" data-language-local-name="vjetnama" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%93%88%E5%AF%86%E9%A1%BF%E5%8A%9B%E5%AD%A6" title="哈密顿力学 — ĉina" lang="zh" hreflang="zh" data-title="哈密顿力学" data-language-autonym="中文" data-language-local-name="ĉina" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q477921#sitelinks-wikipedia" title="Redakti interlingvajn ligilojn" class="wbc-editpage">Redakti ligilojn</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Nomspacoj"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="eo" dir="ltr"><p><b>Hamiltona mekaniko</b> estas reesprimo de <a href="/wiki/Klasika_mekaniko" title="Klasika mekaniko">klasika mekaniko</a> far <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a>. Anstataŭ koordinatoj kaj siaj asociata rapidoj en <a href="/wiki/Lagran%C4%9Da_mekaniko" title="Lagranĝa mekaniko">Lagranĝa mekaniko</a>, Hamiltona mekaniko uzas koordinatoj kaj siaj (kanonaj) <a href="/wiki/Movokvanto" title="Movokvanto">movokvantoj</a>. Tia elekto estas pli "demokratia" en senco ke la koordinatoj kaj la movokvantoj estas reprezentata simile en la ekvacioj de Hamiltona mekaniko (la <b>ekvacioj de Hamilton</b>), kontraste kun la ekvacioj de Euler–Lagrange de <a href="/wiki/Lagran%C4%9Da_mekaniko" title="Lagranĝa mekaniko">Lagranĝa mekaniko</a>. Ankaŭ, la ekvacioj de Hamilton estas unua-ordaj, kontraste kun la dua-ordaj ekvacioj de Euler–Lagrange. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Difino">Difino</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltona_mekaniko&veaction=edit&section=1" title="Redakti sekcion: Difino" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hamiltona_mekaniko&action=edit&section=1" title="Redakti la fontkodo de la sekcio: Difino"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Laŭ hamiltona mekaniko, klasika fizika sistemo konsistas el: </p> <ul><li><a href="/w/index.php?title=Simplekta_sterna%C4%B5o&action=edit&redlink=1" class="new" title="Simplekta sternaĵo (paĝo ne ekzistas)">Simplekta sternaĵo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d343da33a65bc6b0682b8da9ba31e5af966ce9eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.731ex; height:2.843ex;" alt="{\displaystyle (M,\omega )}"></span>, k.e., para-dimensia reela diferenciala <a href="/wiki/Sterna%C4%B5o_(matematiko)" class="mw-redirect" title="Sternaĵo (matematiko)">sternaĵo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> kune kun fermita<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> nedegenera<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Diferenciala_formo" title="Diferenciala formo">diferenciala 2-formo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> (la <b>simplekta formo</b>). La dimensio de <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> estas duobla da la nombro de <a href="/wiki/Grado_de_libereco" title="Grado de libereco">gradoj de libereco</a>. (Pli ĝenerale oni povas uzi <a href="/w/index.php?title=Sterna%C4%B5o_de_Poisson&action=edit&redlink=1" class="new" title="Sternaĵo de Poisson (paĝo ne ekzistas)">sternaĵon de Poisson</a> anstataŭ simplekta sternaĵo.) <b>Stato</b> estas punkto en <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>.</li> <li>Reela funkcio <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\colon \mathbb {R} \times M\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\colon \mathbb {R} \times M\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abafaa717a394c2db0636b9d31e8a8df1ae99cc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.35ex; height:2.176ex;" alt="{\displaystyle H\colon \mathbb {R} \times M\to \mathbb {R} }"></span>, la <b>hamiltoniano</b>, kiu estas funkcio de tempo kaj stato, kaj kies valoro estas (almenaŭ por aŭtonoma sistemo) la <a href="/wiki/Energio" title="Energio">energio</a> de la sistemo. La sistemo estas <b>aŭtonoma</b> <a href="/wiki/S.n.s." class="mw-redirect" title="S.n.s.">s.n.s.</a> la hamiltoniano ne dependas de tempo.</li></ul> <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=T+V,\quad T={\frac {p^{2}}{2m}},\quad V=V(q)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>T</mi> <mo>+</mo> <mi>V</mi> <mo>,</mo> <mspace width="1em" /> <mi>T</mi> <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> <mspace width="1em" /> <mi>V</mi> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=T+V,\quad T={\frac {p^{2}}{2m}},\quad V=V(q)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0200dd6587b30444b02b6537a03178545ead05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.498ex; height:5.676ex;" alt="{\displaystyle H=T+V,\quad T={\frac {p^{2}}{2m}},\quad V=V(q)\,,}"></span></dd></dl> <dl><dd>kie <i>T</i> estas la <a href="/wiki/Kineta_energio" title="Kineta energio">kineta energio</a>, funkcio nur de <a href="/wiki/Movokvanto" title="Movokvanto">movokvanto</a> <i>p</i>, kaj <i>V</i> estas la <a href="/wiki/Potenciala_energio" title="Potenciala energio">potenciala energio</a>, funkcio nur de la koordinato <i>q</i>.</dd></dl> <ul><li>Komenca stato <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 x_{0}\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ffbdb59406dc64aa6769cecf0e9ee109d181119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.667ex; height:2.509ex;" alt="{\displaystyle x_{0}\in M}"></span>.</li></ul> <p>La simplekta formo <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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> difinas izomorfion <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\mapsto \omega (V,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">↦</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\mapsto \omega (V,\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4662fa5c80f60f3b30d10742f669ff918d7d96ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.124ex; height:2.843ex;" alt="{\displaystyle V\mapsto \omega (V,\cdot )}"></span> inter la spaco de vektoroj <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_{x}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{x}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9a02a3b6f9a6808be3b99d0b27d1b97b4bb025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.972ex; height:2.509ex;" alt="{\displaystyle T_{x}M}"></span> kaj la spaco de kovektoroj <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_{x}^{*}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{x}^{*}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4c6a27dfd5f136186c72d25c457ceaba3f467d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.216ex; height:2.509ex;" alt="{\displaystyle T_{x}^{*}M}"></span> ĉe ĉiu punkto <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 x\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df57d73e9532bb93a1439890bcddbc2806f5859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}"></span> — kaj tiel inter vektoraj kampoj kaj 1-formoj (kovektoraj kampoj). Difinu la (2,0)-tensoron <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 \omega ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1653003921b9ecd1be24dd8c9a2505445b64376f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.779ex; height:2.676ex;" alt="{\displaystyle \omega ^{-1}}"></span>. Oni povas do difini la <b>hamiltonan vektoran kampon</b> <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 X_{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d229e5e97360e167654b28a40234485f65027ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.616ex; height:2.509ex;" alt="{\displaystyle X_{H}}"></span> kiel </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 X_{H}(t)=\omega ^{-1}(\operatorname {d} \!H(t),\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{H}(t)=\omega ^{-1}(\operatorname {d} \!H(t),\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1516e4bd9523a9e1421218b708e58629ee2e978b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.637ex; height:3.176ex;" alt="{\displaystyle X_{H}(t)=\omega ^{-1}(\operatorname {d} \!H(t),\cdot )}"></span>.</dd></dl> <p>La stato <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 x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}"></span> evoluas laŭ la <b>ekvacio de Hamilton</b>, kiu asertas ke la evoluo de la stato sekvas la hamiltonan vektoran kampon. Alivorte: </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 {x}}(t)=X_{H}(t,x(t))=\omega ^{-1}(\operatorname {d} \!H(t,x(t)),\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}}(t)=X_{H}(t,x(t))=\omega ^{-1}(\operatorname {d} \!H(t,x(t)),\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8bda42f66edc66e7b8350918b7cbc99609ecbaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.739ex; height:3.176ex;" alt="{\displaystyle {\dot {x}}(t)=X_{H}(t,x(t))=\omega ^{-1}(\operatorname {d} \!H(t,x(t)),\cdot )}"></span>.</dd></dl> <p>Tiu ĉi estas la ekvacio de movado de hamiltona sistemo. </p><p>Loke, oni povas difini lokan koordinatsistemon <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_{i},p_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (q_{i},p_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2645f294d93c48b1e6407b2d7124569d7496ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.649ex; height:2.843ex;" alt="{\displaystyle (q_{i},p_{i})}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dots ,\dim M/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>dim</mi> <mo>⁡</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dots ,\dim M/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd0e425c8716cbf4b2413f87a2c7ace084be5dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.271ex; height:2.843ex;" alt="{\displaystyle i=1,\dots ,\dim M/2}"></span>) tian ke la formo <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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> fariĝas: </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 \omega _{ij}={\begin{pmatrix}0&I_{\dim M/2}\\-I_{\dim M/2}&0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>dim</mi> <mo>⁡</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>dim</mi> <mo>⁡</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{ij}={\begin{pmatrix}0&I_{\dim M/2}\\-I_{\dim M/2}&0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7953ee19b9d14bf9417ecffa71f6da4ec4195a5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.092ex; height:7.509ex;" alt="{\displaystyle \omega _{ij}={\begin{pmatrix}0&I_{\dim M/2}\\-I_{\dim M/2}&0\end{pmatrix}}}"></span></dd></dl> <p>kie <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 I_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aba34f081d776e30204f3458e4f50b403b09e5c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.242ex; height:2.509ex;" alt="{\displaystyle I_{n}}"></span> estas <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 n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> <a href="/wiki/Identa_matrico" title="Identa matrico">identa matrico</a>. Simile, </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 (\omega ^{-1})^{ij}={\begin{pmatrix}0&-I_{\dim M/2}\\I_{\dim M/2}&0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>dim</mi> <mo>⁡</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>dim</mi> <mo>⁡</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\omega ^{-1})^{ij}={\begin{pmatrix}0&-I_{\dim M/2}\\I_{\dim M/2}&0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09f466dd45892106e9908f907b8ae2ef7beda889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.234ex; height:7.509ex;" alt="{\displaystyle (\omega ^{-1})^{ij}={\begin{pmatrix}0&-I_{\dim M/2}\\I_{\dim M/2}&0\end{pmatrix}}}"></span>.</dd></dl> <p>Do la <b>kanonaj ekvacioj de Hamilton</b> fariĝas: </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 {q}}_{i}={\frac {\partial H}{\partial p_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {q}}_{i}={\frac {\partial H}{\partial p_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac1c568c6a4908956f883b10d68b4dfacb94bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.493ex; height:5.843ex;" alt="{\displaystyle {\dot {q}}_{i}={\frac {\partial H}{\partial p_{i}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {p}}_{i}=-{\frac {\partial H}{\partial q_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {p}}_{i}=-{\frac {\partial H}{\partial q_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b934f43614d7c812b707814020ca62235d07de3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:11.373ex; height:5.843ex;" alt="{\displaystyle {\dot {p}}_{i}=-{\frac {\partial H}{\partial q_{i}}}}"></span>.</dd></dl> <p>Ni observu ke la koordinatoj <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2752dcbff884354069fe332b8e51eb0a70a531b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.837ex; height:2.009ex;" alt="{\displaystyle q_{i}}"></span> kaj la movokvantoj <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> estas traktitaj simile (kontraste kun la ekvacioj de Euler–Lagrange de <a href="/wiki/Lagran%C4%9Da_mekaniko" title="Lagranĝa mekaniko">lagranĝa mekaniko</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Krampoj_de_Poisson">Krampoj de Poisson</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltona_mekaniko&veaction=edit&section=2" title="Redakti sekcion: Krampoj de Poisson" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hamiltona_mekaniko&action=edit&section=2" title="Redakti la fontkodo de la sekcio: Krampoj de Poisson"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <b>krampoj de Poisson</b> <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 \{\cdot ,\cdot \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\cdot ,\cdot \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daac56121c7fdfb61a7c33d810be7487e0993460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle \{\cdot ,\cdot \}}"></span> de du skalaraj kampoj <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon M\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon M\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76993355c0ee15a88e300cc419f1b1ef3910b34b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.197ex; height:2.509ex;" alt="{\displaystyle f,g\colon M\to \mathbb {R} }"></span> estas difinitaj kiel </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f,g\}=-\omega ^{-1}(\operatorname {d} \!f,\operatorname {d} \!g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mi>f</mi> <mo>,</mo> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f,g\}=-\omega ^{-1}(\operatorname {d} \!f,\operatorname {d} \!g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b984153399292bb0bb6f95be75ed6e568461e3fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.262ex; height:3.176ex;" alt="{\displaystyle \{f,g\}=-\omega ^{-1}(\operatorname {d} \!f,\operatorname {d} \!g)}"></span>.</dd></dl> <p>Loke, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f,g\}={\frac {\partial f}{\partial q}}{\frac {\partial g}{\partial p}}-{\frac {\partial f}{\partial p}}{\frac {\partial g}{\partial q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>q</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>q</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f,g\}={\frac {\partial f}{\partial q}}{\frac {\partial g}{\partial p}}-{\frac {\partial f}{\partial p}}{\frac {\partial g}{\partial q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/951e8927dd0caa5ba9dba877cadf9846cdffdff5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.152ex; height:6.009ex;" alt="{\displaystyle \{f,g\}={\frac {\partial f}{\partial q}}{\frac {\partial g}{\partial p}}-{\frac {\partial f}{\partial p}}{\frac {\partial g}{\partial q}}}"></span>.</dd></dl> <p>Ilia uzo simpligas la ekvacioj de Hamilton al </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 {q}}_{i}=\{q,H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>i</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>q</mi> <mo>,</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {q}}_{i}=\{q,H\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16cb935265bee7be6e359c99048c112dabd13f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.767ex; height:2.843ex;" alt="{\displaystyle {\dot {q}}_{i}=\{q,H\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {p}}_{i}=\{p,H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>p</mi> <mo>,</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {p}}_{i}=\{p,H\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf8a8afc050d7b2ad8dc38d52acb87b4c11d95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.939ex; height:2.843ex;" alt="{\displaystyle {\dot {p}}_{i}=\{p,H\}}"></span>.</dd></dl> <p>Do la evoluo de ia funkcio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \times M\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \times M\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a200222dd29aff9552d474f23ec3ef7289fc53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.565ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \times M\to \mathbb {R} }"></span> de tempo kaj stato estas </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 {f}}={\frac {\partial f}{\partial q}}{\dot {q}}+{\frac {\partial f}{\partial p}}{\dot {p}}+{\frac {\partial f}{\partial t}}=\{f,H\}+{\frac {\partial f}{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>q</mi> </mrow> </mfrac> </mrow> <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 mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>,</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {f}}={\frac {\partial f}{\partial q}}{\dot {q}}+{\frac {\partial f}{\partial p}}{\dot {p}}+{\frac {\partial f}{\partial t}}=\{f,H\}+{\frac {\partial f}{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9bd0cd7ebf9ad3491889d304405ceb52f3a615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.586ex; height:6.009ex;" alt="{\displaystyle {\dot {f}}={\frac {\partial f}{\partial q}}{\dot {q}}+{\frac {\partial f}{\partial p}}{\dot {p}}+{\frac {\partial f}{\partial t}}=\{f,H\}+{\frac {\partial f}{\partial t}}}"></span>.</dd></dl> <p>Alivorte, ĝenerale, </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 {\frac {\operatorname {d} }{\operatorname {d} \!t}}=\{\cdot ,H\}+{\frac {\partial }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">d</mi> <mrow> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>⋅</mo> <mo>,</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} }{\operatorname {d} \!t}}=\{\cdot ,H\}+{\frac {\partial }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b322eef94182e9440a880f0fb00b8665fa42dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.97ex; height:5.509ex;" alt="{\displaystyle {\frac {\operatorname {d} }{\operatorname {d} \!t}}=\{\cdot ,H\}+{\frac {\partial }{\partial t}}}"></span>.</dd></dl> <p>Ni vidu ke kvanto konserviĝas se ĝiaj krampoj kune kun la hamiltoniano nulas (kaj ĝi ne dependas rekte de tempo). </p> <div class="mw-heading mw-heading2"><h2 id="Principo_de_senmova_ago_por_hamiltonaj_sistemoj">Principo de senmova ago por hamiltonaj sistemoj</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltona_mekaniko&veaction=edit&section=3" title="Redakti sekcion: Principo de senmova ago por hamiltonaj sistemoj" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hamiltona_mekaniko&action=edit&section=3" title="Redakti la fontkodo de la sekcio: Principo de senmova ago por hamiltonaj sistemoj"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Similaĵo al la principo de senmova ago por <a href="/wiki/Lagran%C4%9Da_mekaniko" title="Lagranĝa mekaniko">lagranĝa sistemo</a> ekzistas por hamiltona sistemo. Nomu la spacon de kurboj el <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 x_{0}\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ffbdb59406dc64aa6769cecf0e9ee109d181119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.667ex; height:2.509ex;" alt="{\displaystyle x_{0}\in M}"></span> al <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 x_{1}\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a7a3bbcce8007512fe34453d546b70d953b370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.667ex; height:2.509ex;" alt="{\displaystyle x_{1}\in M}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (x_{0},x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (x_{0},x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39592cdd2572932d299700c724a67aac69e8b9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.064ex; height:2.843ex;" alt="{\displaystyle \Gamma (x_{0},x_{1})}"></span>. Difinu la <b>agon</b> <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\colon \Gamma (x_{0},x_{1})\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>:</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\colon \Gamma (x_{0},x_{1})\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc2bb26748be220e824b8814c97092e2e228801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.889ex; height:2.843ex;" alt="{\displaystyle S\colon \Gamma (x_{0},x_{1})\to \mathbb {R} }"></span> kiel </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[\gamma ]=\int _{t_{0}}^{t_{1}}\left(\sum _{i}p_{i}(t){\dot {q}}_{i}(t)-H(t,q,p)\right)\;\operatorname {d} \!t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <mi>γ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <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>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[\gamma ]=\int _{t_{0}}^{t_{1}}\left(\sum _{i}p_{i}(t){\dot {q}}_{i}(t)-H(t,q,p)\right)\;\operatorname {d} \!t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b75a7d1df82bb4720400933eb7e6b404cd93c17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.554ex; height:7.509ex;" alt="{\displaystyle S[\gamma ]=\int _{t_{0}}^{t_{1}}\left(\sum _{i}p_{i}(t){\dot {q}}_{i}(t)-H(t,q,p)\right)\;\operatorname {d} \!t}"></span>.</dd></dl> <p>Do la ago estas senmova ĉe la trajektorio. Notu ke, por hamiltona sistemo, oni fiksas ambaŭ la koordinatojn kaj la movokvantojn, kontraste kun la principo de senmova ago por lagranĝa sistemo, kie oni fiksas solajn la koordinatojn, ne la rapidojn. </p> <dl><dd><i>Skizo de pruvo</i>.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta S=\delta (\int (p{\dot {q}}-H)\operatorname {d} \!t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>S</mi> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mo>∫</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta S=\delta (\int (p{\dot {q}}-H)\operatorname {d} \!t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b23b02c10921ac7eb0b96a09eb42ef0b474bac38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.572ex; height:5.676ex;" alt="{\displaystyle \delta S=\delta (\int (p{\dot {q}}-H)\operatorname {d} \!t}"></span> <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 =\int \left(\delta p{\dot {q}}+p\delta {\dot {q}}-{\frac {\partial H}{\partial q}}\delta q-{\frac {\partial H}{\partial p}}\delta p\right)\operatorname {d} \!t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo>∫</mo> <mrow> <mo>(</mo> <mrow> <mi>δ</mi> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>p</mi> <mi>δ</mi> <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 mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>q</mi> </mrow> </mfrac> </mrow> <mi>δ</mi> <mi>q</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> </mfrac> </mrow> <mi>δ</mi> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\int \left(\delta p{\dot {q}}+p\delta {\dot {q}}-{\frac {\partial H}{\partial q}}\delta q-{\frac {\partial H}{\partial p}}\delta p\right)\operatorname {d} \!t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/495df57bfaf8999e49d013c4e00a9156f6e39243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.458ex; height:6.176ex;" alt="{\displaystyle =\int \left(\delta p{\dot {q}}+p\delta {\dot {q}}-{\frac {\partial H}{\partial q}}\delta q-{\frac {\partial H}{\partial p}}\delta p\right)\operatorname {d} \!t}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\int \left(\delta p\left({\dot {q}}-{\frac {\partial H}{\partial p}}\right)-\delta q\left({\dot {p}}+{\frac {\partial H}{\partial q}}\right)\right)\operatorname {d} \!t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo>∫</mo> <mrow> <mo>(</mo> <mrow> <mi>δ</mi> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <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 mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>δ</mi> <mi>q</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>q</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\int \left(\delta p\left({\dot {q}}-{\frac {\partial H}{\partial p}}\right)-\delta q\left({\dot {p}}+{\frac {\partial H}{\partial q}}\right)\right)\operatorname {d} \!t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbdd6ad15ba748efb0b43bd1107b63a650511461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.621ex; height:6.176ex;" alt="{\displaystyle =\int \left(\delta p\left({\dot {q}}-{\frac {\partial H}{\partial p}}\right)-\delta q\left({\dot {p}}+{\frac {\partial H}{\partial q}}\right)\right)\operatorname {d} \!t}"></span>.</dd></dl></dd> <dd>∴ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {q}}=\partial H/\partial p}"> <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> <mi mathvariant="normal">∂</mi> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {q}}=\partial H/\partial p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ccab8ad0951771f1c7d31c5e728777fcf8f117" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.507ex; height:2.843ex;" alt="{\displaystyle {\dot {q}}=\partial H/\partial p}"></span> kaj <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 {p}}=-\partial H/\partial q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∂</mi> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {p}}=-\partial H/\partial q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4b09d42944d29907b115052a23c0edf4eb585e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.287ex; height:2.843ex;" alt="{\displaystyle {\dot {p}}=-\partial H/\partial q}"></span> se <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/99aece401b6aa9987a9171cf459d20406bb16a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.809ex; height:2.343ex;" alt="{\displaystyle \delta S=0}"></span> por iu ajn <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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04a2008f8892953aebfa2f8c83545ad7a9e56a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.118ex; height:2.676ex;" alt="{\displaystyle \delta q}"></span> kaj <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2eab305f2f8961b89c0d18722c7cb085f44045f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.218ex; height:2.676ex;" alt="{\displaystyle \delta p}"></span>. ∎</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Teoremo_de_Liouville">Teoremo de Liouville</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltona_mekaniko&veaction=edit&section=4" title="Redakti sekcion: Teoremo de Liouville" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hamiltona_mekaniko&action=edit&section=4" title="Redakti la fontkodo de la sekcio: Teoremo de Liouville"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Natura <a href="/w/index.php?title=Voluma_formo&action=edit&redlink=1" class="new" title="Voluma formo (paĝo ne ekzistas)">voluma formo</a> ekzistas sur simplekta sternaĵo <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,\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d343da33a65bc6b0682b8da9ba31e5af966ce9eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.731ex; height:2.843ex;" alt="{\displaystyle (M,\omega )}"></span>, kiu estas <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 \omega ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cbdd97cf8db5a2f29f648ec7c86cb441d89bfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.664ex; height:2.343ex;" alt="{\displaystyle \omega ^{n}}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim M=2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>M</mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim M=2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c31656c209dd3930200f045eb3aa6137147b059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.36ex; height:2.176ex;" alt="{\displaystyle \dim M=2n}"></span>). Konsideru distribuon (de <a href="/wiki/Ensemblo" class="mw-disambig" title="Ensemblo">ensemblo</a> aŭ <a href="/wiki/Probablo" title="Probablo">probablo</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \colon M\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \colon M\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6db824d60affab23aaba389abcb716ff43d5724" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.97ex; height:2.676ex;" alt="{\displaystyle \rho \colon M\to \mathbb {R} }"></span>. La kvanto <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 N=\int _{M}\rho \omega ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>ρ</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=\int _{M}\rho \omega ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf53625a9f28a1452953e6fc7905e6759e1198ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.667ex; height:5.676ex;" alt="{\displaystyle N=\int _{M}\rho \omega ^{n}}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282a76fe69ce05e31352dfd19b7700eb784fb3f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:2.176ex;" alt="{\displaystyle =1}"></span> por <a href="/wiki/Probablodistribuo" title="Probablodistribuo">probablodistribuo</a>) devas konserviĝi; do la distribuo devas verigi la <a href="/w/index.php?title=Ekvacio_de_kontinueco&action=edit&redlink=1" class="new" title="Ekvacio de kontinueco (paĝo ne ekzistas)">ekvacio de kontinueco</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 \partial _{t}\rho +\partial _{i}(X_{H}^{i}\rho )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>ρ</mi> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{t}\rho +\partial _{i}(X_{H}^{i}\rho )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece6db8ff7c01416090d00fdedc66f1fa32b5db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.025ex; height:3.176ex;" alt="{\displaystyle \partial _{t}\rho +\partial _{i}(X_{H}^{i}\rho )=0}"></span>,</dd></dl> <p>kie <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 X_{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d229e5e97360e167654b28a40234485f65027ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.616ex; height:2.509ex;" alt="{\displaystyle X_{H}}"></span> estas la hamiltona vektora kampo kaj <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 i=1,\dotsc ,\dim M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>dim</mi> <mo>⁡</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dotsc ,\dim M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588afdf04d6ce7ef09869ae2af23f2e40f56ecc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.946ex; height:2.509ex;" alt="{\displaystyle i=1,\dotsc ,\dim M}"></span> estas sumita. Do </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 {\rho }}=\partial _{t}\rho +X_{H}^{i}\partial _{i}\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>ρ</mi> <mo>+</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\rho }}=\partial _{t}\rho +X_{H}^{i}\partial _{i}\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b45a97dfab1919724e9e3fce317966406280ec2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.429ex; height:3.176ex;" alt="{\displaystyle {\dot {\rho }}=\partial _{t}\rho +X_{H}^{i}\partial _{i}\rho }"></span> <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 =-\rho \partial _{i}X_{H}^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo>−</mo> <mi>ρ</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =-\rho \partial _{i}X_{H}^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/812abb61c802307fb5850fc2faea118ec4bc90be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.113ex; height:3.176ex;" alt="{\displaystyle =-\rho \partial _{i}X_{H}^{i}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc9e66de468806365c20e32e83456cc526ce29e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:2.176ex;" alt="{\displaystyle =0}"></span>. (Ĉar <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 _{i}X_{H}^{i}\propto \partial _{i}(\omega ^{ij}\partial _{j}H)=(\partial _{i}\omega ^{ij})\partial _{j}H+\omega ^{ij}\partial _{i}\partial _{j}H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo>∝</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>H</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>H</mi> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{i}X_{H}^{i}\propto \partial _{i}(\omega ^{ij}\partial _{j}H)=(\partial _{i}\omega ^{ij})\partial _{j}H+\omega ^{ij}\partial _{i}\partial _{j}H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d1023777f443565673d3487cd62eefbe683fd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.8ex; height:3.343ex;" alt="{\displaystyle \partial _{i}X_{H}^{i}\propto \partial _{i}(\omega ^{ij}\partial _{j}H)=(\partial _{i}\omega ^{ij})\partial _{j}H+\omega ^{ij}\partial _{i}\partial _{j}H}"></span>; la unua termo nulas ĉar fermiteco de <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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>, la dua ĉar antisimetrio de <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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>.)</dd></dl></dd></dl> <p>Do la probabla denso konserviĝas laŭ hamiltona fluo. Tiu ĉi estas la <b>teoremo de Liouville</b>, pruvita de la usona fizikisto <a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Josiah Willard Gibbs</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> kaj nomita laŭ la franca matematikisto <a href="/w/index.php?title=Joseph_Liouville&action=edit&redlink=1" class="new" title="Joseph Liouville (paĝo ne ekzistas)">Joseph Liouville</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Notoj">Notoj</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltona_mekaniko&veaction=edit&section=5" title="Redakti sekcion: Notoj" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hamiltona_mekaniko&action=edit&section=5" title="Redakti la fontkodo de la sekcio: Notoj"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist references-small" style="padding-right: 6px;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> <span class="reference-text">diferenciala formo <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> estas fermita <a href="/wiki/S.n.s." class="mw-redirect" title="S.n.s.">s.n.s.</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {d} \!\alpha =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {d} \!\alpha =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/120f7304354991837a895b0b9e9385b1c66e8b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.041ex; height:2.176ex;" alt="{\displaystyle \operatorname {d} \!\alpha =0}"></span>.</span> </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> <span class="reference-text">k.e., por ĉiu nenula vektoro <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 X\in T_{x}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∈</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\in T_{x}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a127f93c1cef2774b72b8102126e001961276d12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.793ex; height:2.509ex;" alt="{\displaystyle X\in T_{x}M}"></span> ekzistas vektoro <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 Y\in T_{x}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∈</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\in T_{x}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cdede022c796e2e3509910d87fda41f4e5fbb14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle Y\in T_{x}M}"></span> tia ke <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 \omega (X,Y)\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (X,Y)\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43754e47c161731e0b6759acce313a9183374a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.303ex; height:2.843ex;" alt="{\displaystyle \omega (X,Y)\neq 0}"></span>.</span> </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> <span class="reference-text">JW Gibbs, <i>Elementary principles in statistical mechanics</i> (Elementaj principoj de statistika mekaniko), 1902.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Referencoj">Referencoj</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltona_mekaniko&veaction=edit&section=6" title="Redakti sekcion: Referencoj" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hamiltona_mekaniko&action=edit&section=6" title="Redakti la fontkodo de la sekcio: Referencoj"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Lev_Landau" title="Lev Landau">LD Landau</a>, <a href="/w/index.php?title=Evgeny_Lifshitz&action=edit&redlink=1" class="new" title="Evgeny Lifshitz (paĝo ne ekzistas)">EM Lifshitz</a>, <i>Mechanics</i>, Pergamon Press.</li> <li>KC Gupta, <i>Classical mechanics of particles and rigid bodies</i>, Wiley, 1988.</li> <li>H Goldstein, CP Poole, JL Safko, <i>Classical Mechanics</i>. Addison-Wesley.</li> <li>C Lanczos, <i>The variational principles of mechanics</i>. Dover, 1986, <a href="/wiki/Speciala%C4%B5o:Citoj_el_libroj/0486650677" class="internal mw-magiclink-isbn">ISBN 0486650677</a>.</li> <li>F Kuypers, <i>Klassische Mechanik</i> Wiley-Vch, 2008, <a href="/wiki/Speciala%C4%B5o:Citoj_el_libroj/3527407219" class="internal mw-magiclink-isbn">ISBN 3527407219</a>.</li> <li>ВИ Арнольд, <i>Математические методы классической механики</i>. 3a eld. Moskvo: Наука, 1989. <ul><li><i>Angla traduko</i> VI Arnold, <i>Mathematical methods of mathematical physics</i>, 2a eld. Novjorko: Springer-Verlag, 1989. <a href="/wiki/Speciala%C4%B5o:Citoj_el_libroj/0387968903" class="internal mw-magiclink-isbn">ISBN 0387968903</a></li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Bibliografio">Bibliografio</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltona_mekaniko&veaction=edit&section=7" title="Redakti sekcion: Bibliografio" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hamiltona_mekaniko&action=edit&section=7" title="Redakti la fontkodo de la sekcio: Bibliografio"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Landau & Lifshitz: <i>Mecánica</i>, Ed. Reverté, Barcelono, p.158 - 175, 1991. <a href="/wiki/Speciala%C4%B5o:Citoj_el_libroj/8429140816" class="internal mw-magiclink-isbn">ISBN 84-291-4081-6</a>.(<a href="/wiki/Hispana_lingvo" title="Hispana lingvo"><span style="color:var(--color-subtle, #54595d);">hispane</span></a>)</li> <li>Weisstein, Eric W., "<i><a rel="nofollow" class="external text" href="http://scienceworld.wolfram.com/physics/Hamiltonian.html">Hamiltoniano</a></i>" (<a href="/wiki/Angla_lingvo" title="Angla lingvo"><span style="color:var(--color-subtle, #54595d);">angle</span></a>)</li> <li>Binney, James, "<i><a rel="nofollow" class="external text" href="http://www-thphys.physics.ox.ac.uk/users/JamesBinney/CMech_notes.ps">Klasika mekaniko</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20051025143246/http://www-thphys.physics.ox.ac.uk/users/JamesBinney/CMech_notes.ps">Arkivigite je</a> 2005-10-25 per la retarkivo <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i>" (Postscripta formato <a href="/wiki/PS" class="mw-redirect mw-disambig" title="PS">PS</a>) <a rel="nofollow" class="external text" href="http://www-thphys.physics.ox.ac.uk/users/JamesBinney/cmech.pdf">Notoj de lekcio</a> (Portabla documenta formato <a href="/wiki/PDF" class="mw-redirect" title="PDF">PDF</a>) (<a href="/wiki/Angla_lingvo" title="Angla lingvo"><span style="color:var(--color-subtle, #54595d);">angle</span></a>)</li></ul> <div class="noprint" id="aliaj_projektoj" style="float: right; clear: right; width: 250px; border: 1px solid #aaa; padding: 4px; font-size: 90%; background: #f9f9f9;"> <ul class="noarchive"><li class="commons"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Hamiltonian_mechanics?uselang=eo">Kategorio Hamiltona mekaniko en la <span>Vikimedia Komunejo</span></a> (Multrimedaj datumoj)</span> </li></ul> </div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Elŝutita el "<a dir="ltr" href="https://eo.wikipedia.org/w/index.php?title=Hamiltona_mekaniko&oldid=8725442">https://eo.wikipedia.org/w/index.php?title=Hamiltona_mekaniko&oldid=8725442</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Speciala%C4%B5o:Kategorioj" title="Specialaĵo:Kategorioj">Kategorio</a>: <ul><li><a href="/wiki/Kategorio:Klasika_mekaniko" title="Kategorio:Klasika mekaniko">Klasika mekaniko</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Kaŝitaj kategorioj: <ul><li><a href="/wiki/Kategorio:Listo_de_arkivitaj_pa%C4%9Doj_de_Retarkivo" title="Kategorio:Listo de arkivitaj paĝoj de Retarkivo">Listo de arkivitaj paĝoj de Retarkivo</a></li><li><a href="/wiki/Kategorio:Pa%C4%9Doj_kiuj_uzas_ISBNajn_magiajn_ligilojn" title="Kategorio:Paĝoj kiuj uzas ISBNajn magiajn ligilojn">Paĝoj kiuj uzas ISBNajn magiajn ligilojn</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Ĉi tiu paĝo estis lastafoje redaktita je 16:26, 3 maj. 2024.</li> <li id="footer-info-copyright">La teksto disponeblas laŭ la permesilo <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.eo">Krea Komunaĵo Atribuite-Samkondiĉe 4.0 Neadaptita</a>; eble aldonaj kondiĉoj aplikeblas. 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