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vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contidos</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover á barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">agochar</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inicio</div> </a> </li> <li id="toc-Historia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Historia</span> </div> </a> <button aria-controls="toc-Historia-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>Mostrar ou agochar a subsección "Historia"</span> </button> <ul id="toc-Historia-sublist" class="vector-toc-list"> <li id="toc-Problema_isoperimétrico" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problema_isoperimétrico"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Problema isoperimétrico</span> </div> </a> <ul id="toc-Problema_isoperimétrico-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Braquistócrona" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Braquistócrona"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Braquistócrona</span> </div> </a> <ul id="toc-Braquistócrona-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formulación_xeral" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formulación_xeral"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Formulación xeral</span> </div> </a> <button aria-controls="toc-Formulación_xeral-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>Mostrar ou agochar a subsección "Formulación xeral"</span> </button> <ul id="toc-Formulación_xeral-sublist" class="vector-toc-list"> <li id="toc-Espazos_funcionais" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Espazos_funcionais"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Espazos funcionais</span> </div> </a> <ul id="toc-Espazos_funcionais-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extremos_relativos_débiles_e_fortes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extremos_relativos_débiles_e_fortes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Extremos relativos débiles e fortes</span> </div> </a> <ul id="toc-Extremos_relativos_débiles_e_fortes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Notas</span> </div> </a> <ul id="toc-Notas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Véxase_tamén" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véxase_tamén"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Véxase tamén</span> </div> </a> <button aria-controls="toc-Véxase_tamén-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>Mostrar ou agochar a subsección "Véxase tamén"</span> </button> <ul id="toc-Véxase_tamén-sublist" class="vector-toc-list"> <li id="toc-Bibliografía" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliografía"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Bibliografía</span> </div> </a> <ul id="toc-Bibliografía-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Outros_artigos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Outros_artigos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Outros artigos</span> </div> </a> <ul id="toc-Outros_artigos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligazóns_externas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ligazóns_externas"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Ligazóns externas</span> </div> </a> <ul id="toc-Ligazóns_externas-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="Contidos" 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="Mostrar ou agochar a táboa de contidos" > <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">Mostrar ou agochar a táboa de contidos</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">Cálculo de variacións</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="Ir a un artigo noutra lingua. Dispoñible en 45 linguas" > <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-45" 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">45 linguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%D8%A7%D9%84%D9%85%D8%AA%D8%BA%D9%8A%D8%B1%D8%A7%D8%AA" title="حساب المتغيرات – árabe" lang="ar" hreflang="ar" data-title="حساب المتغيرات" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/C%C3%A1lculu_de_variaciones" title="Cálculu de variaciones – asturiano" lang="ast" hreflang="ast" data-title="Cálculu de variaciones" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D0%B8%D0%B0%D1%86%D0%B8%D0%B0%D0%BB%D1%8B_%D0%B8%D2%AB%D3%99%D0%BF%D0%BB%D3%99%D0%BC%D3%99" title="Вариациалы иҫәпләмә – baxkir" lang="ba" hreflang="ba" data-title="Вариациалы иҫәпләмә" data-language-autonym="Башҡортса" data-language-local-name="baxkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D1%8B%D1%8F%D1%86%D1%8B%D0%B9%D0%BD%D0%B0%D0%B5_%D0%B7%D0%BB%D1%96%D1%87%D1%8D%D0%BD%D0%BD%D0%B5" title="Варыяцыйнае злічэнне – belaruso" lang="be" hreflang="be" data-title="Варыяцыйнае злічэнне" data-language-autonym="Беларуская" data-language-local-name="belaruso" 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%92%D0%B0%D1%80%D0%B8%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%BE_%D1%81%D0%BC%D1%8F%D1%82%D0%B0%D0%BD%D0%B5" title="Вариационно смятане – búlgaro" lang="bg" hreflang="bg" data-title="Вариационно смятане" data-language-autonym="Български" data-language-local-name="búlgaro" 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/C%C3%A0lcul_de_variacions" title="Càlcul de variacions – catalán" lang="ca" hreflang="ca" data-title="Càlcul de variacions" data-language-autonym="Català" data-language-local-name="catalán" 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/Varia%C4%8Dn%C3%AD_po%C4%8Det" title="Variační počet – checo" lang="cs" hreflang="cs" data-title="Variační počet" data-language-autonym="Čeština" data-language-local-name="checo" 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%92%D0%B0%D1%80%D0%B8%D0%B0%D1%86%D0%B8%D0%BB%D0%BB%D0%B5_%D1%88%D1%83%D1%82%D0%BB%D0%B0%D0%B2" title="Вариацилле шутлав – chuvaxo" lang="cv" hreflang="cv" data-title="Вариацилле шутлав" data-language-autonym="Чӑвашла" data-language-local-name="chuvaxo" 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/Variationsrechnung" title="Variationsrechnung – alemán" lang="de" hreflang="de" data-title="Variationsrechnung" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9B%CE%BF%CE%B3%CE%B9%CF%83%CE%BC%CF%8C%CF%82_%CF%84%CF%89%CE%BD_%CE%BC%CE%B5%CF%84%CE%B1%CE%B2%CE%BF%CE%BB%CF%8E%CE%BD" title="Λογισμός των μεταβολών – grego" lang="el" hreflang="el" data-title="Λογισμός των μεταβολών" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Calculus_of_variations" title="Calculus of variations – inglés" lang="en" hreflang="en" data-title="Calculus of variations" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Variada_kalkulo" title="Variada kalkulo – esperanto" lang="eo" hreflang="eo" data-title="Variada kalkulo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/C%C3%A1lculo_de_variaciones" title="Cálculo de variaciones – español" lang="es" hreflang="es" data-title="Cálculo de variaciones" data-language-autonym="Español" data-language-local-name="español" 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/Variatsioonarvutus" title="Variatsioonarvutus – estoniano" lang="et" hreflang="et" data-title="Variatsioonarvutus" data-language-autonym="Eesti" data-language-local-name="estoniano" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bariazioen_kalkulu" title="Bariazioen kalkulu – éuscaro" lang="eu" hreflang="eu" data-title="Bariazioen kalkulu" data-language-autonym="Euskara" data-language-local-name="éuscaro" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%D8%AA%D8%BA%DB%8C%DB%8C%D8%B1%D8%A7%D8%AA" title="حساب تغییرات – persa" lang="fa" hreflang="fa" data-title="حساب تغییرات" 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/Variaatiolaskenta" title="Variaatiolaskenta – finés" lang="fi" hreflang="fi" data-title="Variaatiolaskenta" data-language-autonym="Suomi" data-language-local-name="finés" 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/Calcul_des_variations" title="Calcul des variations – francés" lang="fr" hreflang="fr" data-title="Calcul des variations" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%95%D7%A8%D7%99%D7%90%D7%A6%D7%99%D7%95%D7%AA" title="חשבון וריאציות – hebreo" lang="he" hreflang="he" data-title="חשבון וריאציות" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%9A%E0%A4%B0%E0%A4%A3-%E0%A4%95%E0%A4%B2%E0%A4%A8" title="विचरण-कलन – hindi" lang="hi" hreflang="hi" data-title="विचरण-कलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Vari%C3%A1ci%C3%B3sz%C3%A1m%C3%ADt%C3%A1s" title="Variációszámítás – húngaro" lang="hu" hreflang="hu" data-title="Variációszámítás" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Calcolo_delle_variazioni" title="Calcolo delle variazioni – italiano" lang="it" hreflang="it" data-title="Calcolo delle variazioni" data-language-autonym="Italiano" data-language-local-name="italiano" 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/%E5%A4%89%E5%88%86%E6%B3%95" title="変分法 – xaponés" lang="ja" hreflang="ja" data-title="変分法" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D0%B8%D0%B0%D1%86%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D0%B5%D1%81%D0%B5%D0%BF%D1%82%D0%B5%D1%83" title="Вариациялық есептеу – kazako" lang="kk" hreflang="kk" data-title="Вариациялық есептеу" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%80%EB%B6%84%EB%B2%95" title="변분법 – coreano" lang="ko" hreflang="ko" data-title="변분법" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Kalkulu_tal-varjazzjonijiet" title="Kalkulu tal-varjazzjonijiet – maltés" lang="mt" hreflang="mt" data-title="Kalkulu tal-varjazzjonijiet" data-language-autonym="Malti" data-language-local-name="maltés" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Variatierekening" title="Variatierekening – neerlandés" lang="nl" hreflang="nl" data-title="Variatierekening" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Variasjonsrekning" title="Variasjonsrekning – noruegués nynorsk" lang="nn" hreflang="nn" data-title="Variasjonsrekning" data-language-autonym="Norsk nynorsk" data-language-local-name="noruegués nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Variasjonsregning" title="Variasjonsregning – noruegués bokmål" lang="nb" hreflang="nb" data-title="Variasjonsregning" data-language-autonym="Norsk bokmål" data-language-local-name="noruegués bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rachunek_wariacyjny" title="Rachunek wariacyjny – polaco" lang="pl" hreflang="pl" data-title="Rachunek wariacyjny" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/C%C3%A0lcol_dle_variassion" title="Càlcol dle variassion – Piedmontese" lang="pms" hreflang="pms" data-title="Càlcol dle variassion" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/C%C3%A1lculo_variacional" title="Cálculo variacional – portugués" lang="pt" hreflang="pt" data-title="Cálculo variacional" data-language-autonym="Português" data-language-local-name="portugués" 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/Calcul_varia%C8%9Bional" title="Calcul variațional – romanés" lang="ro" hreflang="ro" data-title="Calcul variațional" data-language-autonym="Română" data-language-local-name="romanés" 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%92%D0%B0%D1%80%D0%B8%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%BE%D0%B5_%D0%B8%D1%81%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Вариационное исчисление – ruso" lang="ru" hreflang="ru" data-title="Вариационное исчисление" data-language-autonym="Русский" data-language-local-name="ruso" 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/Calculus_of_variations" title="Calculus of variations – Simple English" lang="en-simple" hreflang="en-simple" data-title="Calculus of variations" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Varia%C4%8Dn%C3%BD_po%C4%8Det" title="Variačný počet – eslovaco" lang="sk" hreflang="sk" data-title="Variačný počet" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Variacijski_ra%C4%8Dun" title="Variacijski račun – esloveno" lang="sl" hreflang="sl" data-title="Variacijski račun" data-language-autonym="Slovenščina" data-language-local-name="esloveno" 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/Analiza_e_variacionit" title="Analiza e variacionit – albanés" lang="sq" hreflang="sq" data-title="Analiza e variacionit" data-language-autonym="Shqip" data-language-local-name="albanés" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Varijacijski_ra%C4%8Dun" title="Varijacijski račun – serbio" lang="sr" hreflang="sr" data-title="Varijacijski račun" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Variationskalkyl" title="Variationskalkyl – sueco" lang="sv" hreflang="sv" data-title="Variationskalkyl" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D1%96%D0%B0%D1%86%D1%96%D0%B9%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Варіаційне числення – ucraíno" lang="uk" hreflang="uk" data-title="Варіаційне числення" 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class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Ferramentas das páxinas"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparencia"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aparencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover á barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">agochar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Na Galipedia, a Wikipedia en galego.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="gl" dir="ltr"><p>O <b>cálculo de variacións</b> é un problema matemático consistente en buscar <a href="/wiki/Extremos_dunha_funci%C3%B3n" title="Extremos dunha función">máximos e mínimos</a> (ou máis xeralmente extremos relativos) de <a href="/w/index.php?title=Funcional&action=edit&redlink=1" class="new" title="Funcional (a páxina aínda non existe)">funcionais</a> continuos definidos sobre algún espazo funcional. </p><p>Constitúen unha xeneralización do cálculo elemental de máximos e mínimos de <a href="/w/index.php?title=Funci%C3%B3n_real&action=edit&redlink=1" class="new" title="Función real (a páxina aínda non existe)">funcións reais</a> dunha variable </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&veaction=edit&section=1" title="Editar a sección: «Historia»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&action=edit&section=1" title="Editar o código fonte da sección: Historia"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O cálculo de variacións desenvolveuse a partir do problema da <a href="/w/index.php?title=Curva_braquist%C3%B3crona&action=edit&redlink=1" class="new" title="Curva braquistócrona (a páxina aínda non existe)">curva braquistócrona</a>, exposto inicialmente por <a href="/w/index.php?title=Johann_Bernoulli&action=edit&redlink=1" class="new" title="Johann Bernoulli (a páxina aínda non existe)">Johann Bernoulli</a> (1696). Inmediatamente este problema captou a atención de <a href="/wiki/Jakob_Bernoulli" title="Jakob Bernoulli">Jakob Bernoulli</a> e o <a href="/wiki/Guillaume_de_l%27H%C3%B4pital" title="Guillaume de l'Hôpital">marqués de L'Hôpital</a>, aínda que foi <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> o primeiro que elaborou unha teoría do cálculo variacional. As contribucións de Euler iniciáronse en 1733 coa súa <i>Elementa Calculi Variationum</i> ("Elementos do cálculo de variacións") que deu nome á disciplina. </p><p><a href="/wiki/Joseph_Louis_Lagrange" title="Joseph Louis Lagrange">Lagrange</a> contribuíu cumpridamente á teoría e <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Legendre</a> (1786) asentou un método, non enteiramente satisfactorio para distinguir entre máximos e mínimos. <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> e <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Leibniz</a> tamén prestaron atención a este asunto. Outros traballos destacados foron os de <a href="/w/index.php?title=Vincenzo_Brunacci&action=edit&redlink=1" class="new" title="Vincenzo Brunacci (a páxina aínda non existe)">Vincenzo Brunacci</a> (1810), <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (1829), <a href="/wiki/Sim%C3%A9on_Poisson" class="mw-redirect" title="Siméon Poisson">Siméon Poisson</a> (1831), <a href="/w/index.php?title=Mija%C3%ADl_Ostrogradski&action=edit&redlink=1" class="new" title="Mijaíl Ostrogradski (a páxina aínda non existe)">Mikhail Ostrogradski</a> (1834) e <a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Carl Jacobi</a> (1837). Un traballo xeral particularmente importante é o de <a href="/w/index.php?title=Pierre_Fr%C3%A9d%C3%A9ric_Sarrus&action=edit&redlink=1" class="new" title="Pierre Frédéric Sarrus (a páxina aínda non existe)">Sarrus</a> (1842) que foi resumido por <a href="/wiki/Augustin_Louis_Cauchy" title="Augustin Louis Cauchy">Cauchy</a> (1844). Outros traballos destacados posteriores son os de <a href="/w/index.php?title=Strauch&action=edit&redlink=1" class="new" title="Strauch (a páxina aínda non existe)">Strauch</a> (1849), <a href="/w/index.php?title=Jellett&action=edit&redlink=1" class="new" title="Jellett (a páxina aínda non existe)">Jellett</a> (1850), <a href="/w/index.php?title=Otto_Hesse&action=edit&redlink=1" class="new" title="Otto Hesse (a páxina aínda non existe)">Otto Hesse</a> (1857), <a href="/w/index.php?title=Alfred_Clebsch&action=edit&redlink=1" class="new" title="Alfred Clebsch (a páxina aínda non existe)">Alfred Clebsch</a> (1858) e <a href="/w/index.php?title=Carll&action=edit&redlink=1" class="new" title="Carll (a páxina aínda non existe)">Carll</a> (1885), aínda que quizais o máis importante dos traballos durante o século XIX é o de <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Weierstrass.</a> Este importante traballo foi unha referencia estándar e é o primeiro que trata o cálculo de variacións sobre unha base firme e rigorosa. Os <a href="/wiki/Problemas_de_Hilbert" title="Problemas de Hilbert">problemas 20 e 23 de Hilbert</a> expostos en 1900 estimularon algúns desenvolvementos posteriores. Durante o século XX, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>, <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a>, <a href="/w/index.php?title=Leonida_Tonelli&action=edit&redlink=1" class="new" title="Leonida Tonelli (a páxina aínda non existe)">Leonida Tonelli</a>, <a href="/wiki/Henri_L%C3%A9on_Lebesgue" title="Henri Léon Lebesgue">Henri Lebesgue</a> e <a href="/wiki/Jacques_Hadamard" title="Jacques Hadamard">Jacques Hadamard</a>, entre outros, fixeron contribucións notables. <a href="/w/index.php?title=Marston_Morse&action=edit&redlink=1" class="new" title="Marston Morse (a páxina aínda non existe)">Marston Morse</a> aplicou o cálculo de variacións ao que actualmente se coñece como <a href="/w/index.php?title=Teor%C3%ADa_de_Morse&action=edit&redlink=1" class="new" title="Teoría de Morse (a páxina aínda non existe)">teoría de Morse</a>. <a href="/w/index.php?title=Lev_Semenovich_Pontryagin&action=edit&redlink=1" class="new" title="Lev Semenovich Pontryagin (a páxina aínda non existe)">Lev Semenovich Pontryagin</a>, <a href="/w/index.php?title=R._Tyrrell_Rockafellar&action=edit&redlink=1" class="new" title="R. Tyrrell Rockafellar (a páxina aínda non existe)">Ralph Rockafellar</a> e Clarke desenvolveron novas ferramentas matemáticas dentro da teoría do control óptimo, xeneralizando o cálculo de variacións. </p> <div class="mw-heading mw-heading3"><h3 id="Problema_isoperimétrico"><span id="Problema_isoperim.C3.A9trico"></span>Problema isoperimétrico</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&veaction=edit&section=2" title="Editar a sección: «Problema isoperimétrico»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&action=edit&section=2" title="Editar o código fonte da sección: Problema isoperimétrico"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cal é a área máxima <i>A</i> que pode rodearse cunha curva de lonxitude <i>L</i> dada? De non existiren restricións adicionais, a solución é: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><br /> <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 A={\frac {L^{2}}{4\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {L^{2}}{4\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c23d4e806cda78da19ca27890bd4228de6f29d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.315ex; height:5.676ex;" alt="{\displaystyle A={\frac {L^{2}}{4\pi }}}"></span> </p> </blockquote> <p>que é o valor que se obtén para un <a href="/wiki/C%C3%ADrculo" title="Círculo">círculo</a> de <a href="/wiki/Raio_(xeometr%C3%ADa)" title="Raio (xeometría)">raio</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 R=L/2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=L/2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55261c78426504109e223e19e431fc5253030e1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.102ex; height:2.843ex;" alt="{\displaystyle R=L/2\pi }"></span>. </p><p>Se se impoñen restricións adicionais a solución é diferente. Un exemplo é se se supón que <i>L</i> se considera sobre unha <a href="/wiki/Funci%C3%B3n" title="Función">función</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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> e os extremos da curva están sobre os puntos <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 A=(a,0),B=(b,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(a,0),B=(b,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71274de17dbb4f75571e809604cafd96bdfd8c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.977ex; height:2.843ex;" alt="{\displaystyle A=(a,0),B=(b,0)}"></span> onde a distancia entre eles está dada. É dicir <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 AB=L\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mi>L</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB=L\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369c425dcbc66cf656bc9b3da5a06da0c7ac4c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.576ex; height:2.176ex;" alt="{\displaystyle AB=L\,}"></span>. O problema de achar unha curva que maximice a área entre ela e o eixe <i>X</i> sería atopar unha función <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> de modo que: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><br /> <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 \max _{f:[a,b]\to \mathbb {R} }I[f]=\int _{a}^{b}f(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </munder> <mi>I</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max _{f:[a,b]\to \mathbb {R} }I[f]=\int _{a}^{b}f(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0edbd94617f1172835c275c10031436e1282bc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.12ex; height:6.509ex;" alt="{\displaystyle \max _{f:[a,b]\to \mathbb {R} }I[f]=\int _{a}^{b}f(x)dx}"></span> </p> </blockquote> <p>coas restricións: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}G[f]=\int _{a}^{b}{\sqrt {1+(f'(x))^{2}}}dx=L&{\mbox{longitud de arco}}\\f(a)=f(b)=0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>G</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>L</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>longitud de arco</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}G[f]=\int _{a}^{b}{\sqrt {1+(f'(x))^{2}}}dx=L&{\mbox{longitud de arco}}\\f(a)=f(b)=0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07e76281535e5edddf09b356822c5ee9cbed6131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.871ex; height:6.509ex;" alt="{\displaystyle {\begin{cases}G[f]=\int _{a}^{b}{\sqrt {1+(f'(x))^{2}}}dx=L&{\mbox{longitud de arco}}\\f(a)=f(b)=0\end{cases}}}"></span> </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Braquistócrona"><span id="Braquist.C3.B3crona"></span>Braquistócrona</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&veaction=edit&section=3" title="Editar a sección: «Braquistócrona»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&action=edit&section=3" title="Editar o código fonte da sección: Braquistócrona"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O problema da curva <a href="/w/index.php?title=Braquist%C3%B3crona&action=edit&redlink=1" class="new" title="Braquistócrona (a páxina aínda non existe)">braquistócrona</a> remóntase a <a href="/wiki/Jakob_Bernoulli" title="Jakob Bernoulli">Jakob Bernoulli</a> (1696). Refírese a atopar unha curva no plano cartesiano que vaia do punto <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=(x_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(x_{0},y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a29b6070ac1b7ca23c822ac47b0b1bf4593ab526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.265ex; height:2.843ex;" alt="{\displaystyle P=(x_{0},y_{0})}"></span> á orixe de modo que un punto material que se desliza sen fricción sobre ela tarda o menor tempo posible en ir 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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> á orixe. Usando principios de <a href="/wiki/Mec%C3%A1nica_cl%C3%A1sica" title="Mecánica clásica">mecánica clásica</a> o problema pode formularse como, </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><br /> <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 \min _{f}T[f]=\int _{0}^{x_{0}}{\frac {\sqrt {1+(f'(x))^{2}}}{\sqrt {2g(y_{0}-y)}}}\ dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </munder> <mi>T</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <msqrt> <mn>2</mn> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mtext> </mtext> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min _{f}T[f]=\int _{0}^{x_{0}}{\frac {\sqrt {1+(f'(x))^{2}}}{\sqrt {2g(y_{0}-y)}}}\ dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b1d04f1f08b5584f4af926fb849bb2030ff36f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.721ex; height:7.509ex;" alt="{\displaystyle \min _{f}T[f]=\int _{0}^{x_{0}}{\frac {\sqrt {1+(f'(x))^{2}}}{\sqrt {2g(y_{0}-y)}}}\ dx}"></span> </p> </blockquote> <p>onde <i>g</i> é a <a href="/wiki/Gravidade" title="Gravidade">gravidade</a> e as restricións son, <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(0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d308c32c9894b88115262081194321ae7d9bbf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=0}"></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 f(x_{0})=y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{0})=y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1661f200e31065bfb7291ec00a2e829cfe18f97d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.764ex; height:2.843ex;" alt="{\displaystyle f(x_{0})=y_{0}}"></span>. Hai que notar que 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 x=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e899fc6eba0b387b91f070adc7bc4fe5a706cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{0}}"></span> existe unha <a href="/w/index.php?title=Singularidade_matem%C3%A1tica&action=edit&redlink=1" class="new" title="Singularidade matemática (a páxina aínda non existe)">singularidade</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Formulación_xeral"><span id="Formulaci.C3.B3n_xeral"></span>Formulación xeral</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&veaction=edit&section=4" title="Editar a sección: «Formulación xeral»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&action=edit&section=4" title="Editar o código fonte da sección: Formulación xeral"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un dos problemas típicos en <a href="/wiki/C%C3%A1lculo_diferencial" title="Cálculo diferencial">cálculo diferencial</a> é o de atopar o valor 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 x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab34739435d9d9d99cddf4041740b107343b1398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle x\,}"></span> para o cal a función <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(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78b2b66021c2cac2b5654495678c63ff142952e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.805ex; height:2.843ex;" alt="{\displaystyle f(x)\,}"></span> alcanza un valor extremo (máximo ou mínimo). No cálculo de variacións o problema é atopar unha <a href="/wiki/Funci%C3%B3n" title="Función">función</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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> para a cal un funcional <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 J[f]\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J[f]\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ea0bf8b1b1e72b887230a17788d39eb3ba7fd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.431ex; height:2.843ex;" alt="{\displaystyle J[f]\,}"></span> alcance un valor extremo. O funcional <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 J[f]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J[f]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de95b8fa7148a0649ae7783e6d6d0fb67ab5e18f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.044ex; height:2.843ex;" alt="{\displaystyle J[f]}"></span> está composto por unha <a href="/wiki/Integral" title="Integral">integral</a> que depende 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, da función <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> e algunhas das súas <a href="/wiki/Derivada" title="Derivada">derivadas</a>. </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><span style="float:right; width:10%; text-align:right;">(<cite id="Equation_1a" style="font-style: normal;"><a href="#Eqnref_1a">1a</a></cite>)</span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max _{f}/\min _{f}\left\{I[f]=\int _{a}^{b}{\mathcal {L}}(x,f(x),f'(x),f''(x),\dots ,f^{(n}(x))\,dx\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </munder> <mrow> <mo>{</mo> <mrow> <mi>I</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max _{f}/\min _{f}\left\{I[f]=\int _{a}^{b}{\mathcal {L}}(x,f(x),f'(x),f''(x),\dots ,f^{(n}(x))\,dx\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a80931d48f42382bd8b4dc9f25a16bf0d9337241" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.074ex; height:6.509ex;" alt="{\displaystyle \max _{f}/\min _{f}\left\{I[f]=\int _{a}^{b}{\mathcal {L}}(x,f(x),f'(x),f''(x),\dots ,f^{(n}(x))\,dx\right\}}"></span> </p> </blockquote> <p>Onde a función <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> pertence a algún espazo de funcións (<a href="/w/index.php?title=Espazo_de_Banach&action=edit&redlink=1" class="new" title="Espazo de Banach (a páxina aínda non existe)">espazo de Banach</a>, <a href="/wiki/Espazo_de_Hilbert" title="Espazo de Hilbert">espazo de Hilbert</a>), e tanto ela como as súas derivadas poden ter restricións. Esta fórmula integral pode ser máis complicada permitindo 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> ser un <a href="/wiki/Vector" title="Vector">vector</a>, e polo tanto incluíndo derivadas parciais para <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><span style="float:right; width:10%; text-align:right;">(<cite id="Equation_1b" style="font-style: normal;"><a href="#Eqnref_1b">1b</a></cite>)</span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max _{\mathbf {f} }/\min _{\mathbf {f} }\left\{J[\mathbf {f} ]=\int _{{\mathcal {D}}\subset \mathbb {R} ^{n}}{\mathcal {L}}(\mathbf {x} ,\mathbf {f} (\mathbf {x} ),D\mathbf {f} (\mathbf {x} ),\dots ,D^{n}\mathbf {f} (\mathbf {x} ))\,d^{n}\mathbf {x} \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mrow> </munder> <mrow> <mo>{</mo> <mrow> <mi>J</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max _{\mathbf {f} }/\min _{\mathbf {f} }\left\{J[\mathbf {f} ]=\int _{{\mathcal {D}}\subset \mathbb {R} ^{n}}{\mathcal {L}}(\mathbf {x} ,\mathbf {f} (\mathbf {x} ),D\mathbf {f} (\mathbf {x} ),\dots ,D^{n}\mathbf {f} (\mathbf {x} ))\,d^{n}\mathbf {x} \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7653ed90cdff2c9c5c416f6600426cfb095d0451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.729ex; height:6.176ex;" alt="{\displaystyle \max _{\mathbf {f} }/\min _{\mathbf {f} }\left\{J[\mathbf {f} ]=\int _{{\mathcal {D}}\subset \mathbb {R} ^{n}}{\mathcal {L}}(\mathbf {x} ,\mathbf {f} (\mathbf {x} ),D\mathbf {f} (\mathbf {x} ),\dots ,D^{n}\mathbf {f} (\mathbf {x} ))\,d^{n}\mathbf {x} \right\}}"></span> </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Espazos_funcionais">Espazos funcionais</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&veaction=edit&section=5" title="Editar a sección: «Espazos funcionais»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&action=edit&section=5" title="Editar o código fonte da sección: Espazos funcionais"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A fundamentación rigorosa do cálculo de variacións require considerar variedades diferenciais lineares de <a href="/w/index.php?title=Dimensi%C3%B3n_infinita&action=edit&redlink=1" class="new" title="Dimensión infinita (a páxina aínda non existe)">dimensión infinita</a>. De feito o punto de partida do cálculo de variacións é un teorema da análise funcional que proba que é posible considerar unha curva nun espazo funcional (p.ex. traxectoria no <a href="/w/index.php?title=Espazo_f%C3%A1sico&action=edit&redlink=1" class="new" title="Espazo fásico (a páxina aínda non existe)">espazo fásico</a>) simplemente como unha función cunha variable adicional, concretamente:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup> <style data-mw-deduplicate="TemplateStyles:r6856698">.mw-parser-output .flexquote{display:flex;flex-direction:column;background:var(--background-color-neutral-subtle,#f8f9fa);border-left:3px solid var(--border-color-subtle,#c8ccd1);font-size:90%;margin:1em 4em;padding:.4em .8em}.mw-parser-output .flexquote>.flex{display:flex;flex-direction:row}.mw-parser-output .flexquote>.flex>.quote{width:100%}.mw-parser-output .flexquote>.flex>.separator{border-left:1px solid var(--border-color-subtle,#c8ccd1);border-top:1px solid var(--border-color-subtle,#c8ccd1);margin:.4em .8em}.mw-parser-output .flexquote>.cite{text-align:right}@media all and (max-width:600px){.mw-parser-output .flexquote>.flex{flex-direction:column}}</style> </p> <blockquote class="flexquote"> <div class="flex"> <div class="quote">A categoría formada por <a href="/w/index.php?title=Espazo_vectorial_conveniente&action=edit&redlink=1" class="new" title="Espazo vectorial conveniente (a páxina aínda non existe)">espazos vectoriais convenientes</a> e <a href="/w/index.php?title=Funci%C3%B3n_suave&action=edit&redlink=1" class="new" title="Función suave (a páxina aínda non existe)">funcións suaves</a> entre eles é pechada polo produto cartesiano, de tal manera que se ten a seguinte bixección natural:<i></i> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><br /> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(E\times F,G)\approx C^{\infty }(E,C^{\infty }(F,G))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>E</mi> <mo>×</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(E\times F,G)\approx C^{\infty }(E,C^{\infty }(F,G))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/def4c066e45297643b34a39502b370ecb0897a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.174ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(E\times F,G)\approx C^{\infty }(E,C^{\infty }(F,G))}"></span></dd></dl> </blockquote> onde <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 \scriptstyle E,F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>E</mi> <mo>,</mo> <mi>F</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle E,F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f100a740364690189ed4db7a5d278c02c174881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.944ex; height:1.843ex;" alt="{\displaystyle \scriptstyle E,F}"></span> son espazos vectoriais convenientes e a bixección anterior é un difeomorfismo.</div> </div> </blockquote> <p>O teorema anterior pode aplicarse por exemplo ao <a href="/w/index.php?title=Principio_de_m%C3%ADnima_acci%C3%B3n&action=edit&redlink=1" class="new" title="Principio de mínima acción (a páxina aínda non existe)">principio de mínima acción</a> onde trata de atoparse a traxectoria posible no espazo de fases que fai mínima a integral de acción. Dita traxectoria é unha curva suave no espazo de traxectorias <i>E</i>, considerando agora: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=C^{\infty }(\mathbb {R} ,\mathbb {R} ^{n}),\quad F=G=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>F</mi> <mo>=</mo> <mi>G</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=C^{\infty }(\mathbb {R} ,\mathbb {R} ^{n}),\quad F=G=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24aa9e48438f01e29a650bd2bb8b67c56281f9c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.764ex; height:2.843ex;" alt="{\displaystyle E=C^{\infty }(\mathbb {R} ,\mathbb {R} ^{n}),\quad F=G=\mathbb {R} }"></span> </p> </blockquote> <p>Tense que o problema de minimización pode reducirse a minimizar unha certa función real <i>f</i> de variable real: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><br /> <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_{q_{0}}(\varepsilon ):=S[q_{0}+\varepsilon \delta q],\qquad S:\mathbb {C} ^{\infty }(\mathbb {R} ^{n})\to \mathbb {R} ,\ S[q]:=\int _{t_{1}}^{t_{2}}{\mathcal {L}}(q(t),{\dot {q}}(t),t)\ dt,\ q(t)\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>S</mi> <mo stretchy="false">[</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mi>δ</mi> <mi>q</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mspace width="2em" /> <mi>S</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>S</mi> <mo stretchy="false">[</mo> <mi>q</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>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mo>,</mo> <mtext> </mtext> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{q_{0}}(\varepsilon ):=S[q_{0}+\varepsilon \delta q],\qquad S:\mathbb {C} ^{\infty }(\mathbb {R} ^{n})\to \mathbb {R} ,\ S[q]:=\int _{t_{1}}^{t_{2}}{\mathcal {L}}(q(t),{\dot {q}}(t),t)\ dt,\ q(t)\in \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/849760f0cfd3f0afa338b070d6fad2f405858011" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:84.67ex; height:6.509ex;" alt="{\displaystyle f_{q_{0}}(\varepsilon ):=S[q_{0}+\varepsilon \delta q],\qquad S:\mathbb {C} ^{\infty }(\mathbb {R} ^{n})\to \mathbb {R} ,\ S[q]:=\int _{t_{1}}^{t_{2}}{\mathcal {L}}(q(t),{\dot {q}}(t),t)\ dt,\ q(t)\in \mathbb {R} ^{n}}"></span> </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Extremos_relativos_débiles_e_fortes"><span id="Extremos_relativos_d.C3.A9biles_e_fortes"></span>Extremos relativos débiles e fortes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&veaction=edit&section=6" title="Editar a sección: «Extremos relativos débiles e fortes»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&action=edit&section=6" title="Editar o código fonte da sección: Extremos relativos débiles e fortes"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un problema variacional require que o funcional <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 \scriptstyle J(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle J(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc5774343635f21b135989a7293f6cfa59f4a35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.224ex; height:2.176ex;" alt="{\displaystyle \scriptstyle J(f)}"></span> estea definido sobre un <a href="/w/index.php?title=Espazo_de_Banach&action=edit&redlink=1" class="new" title="Espazo de Banach (a páxina aínda non existe)">espazo de Banach</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 \scriptstyle (V,\|\cdot \|_{V})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle (V,\|\cdot \|_{V})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0acea78c75183c72778ae5939a475906d20bb93d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.292ex; height:2.343ex;" alt="{\displaystyle \scriptstyle (V,\|\cdot \|_{V})}"></span> adecuado. A <a href="/wiki/Norma_(matem%C3%A1ticas)" title="Norma (matemáticas)">norma vectorial</a> de devandito espazo é o que permite definir rigorosamente se unha solución é un mínimo ou un máximo relativo. Por exemplo unha función <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 \scriptstyle f_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle f_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4885ad35ba3bb0582fffe651fd23170c13179d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.637ex; height:2.009ex;" alt="{\displaystyle \scriptstyle f_{0}}"></span> é un mínimo relativo se existe un certo <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 \scriptstyle \delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7307c8f50fe1fbd19076bd7b3292a93b81046e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.842ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \delta >0}"></span> tal que, para toda función <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 \scriptstyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>f</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb782e08eca6897b057d996121da1dbbc94a6f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:0.904ex; height:2.009ex;" alt="{\displaystyle \scriptstyle f}"></span> se cumpre que: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f-f_{0}\|<\delta \quad \Rightarrow \quad J(f_{0})\leq J(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>−</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mi>δ</mi> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <mi>J</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>J</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f-f_{0}\|<\delta \quad \Rightarrow \quad J(f_{0})\leq J(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94e87b4239f5987cdb93b183f7b7ea2f10d62aba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.176ex; height:2.843ex;" alt="{\displaystyle \|f-f_{0}\|<\delta \quad \Rightarrow \quad J(f_{0})\leq J(f)}"></span> </p> </blockquote> <div class="mw-heading mw-heading2"><h2 id="Notas">Notas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&veaction=edit&section=7" title="Editar a sección: «Notas»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&action=edit&section=7" title="Editar o código fonte da sección: Notas"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">A. Kriegl y P. Michor, 1989, p. 3</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Véxase_tamén"><span id="V.C3.A9xase_tam.C3.A9n"></span>Véxase tamén</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&veaction=edit&section=8" title="Editar a sección: «Véxase tamén»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&action=edit&section=8" title="Editar o código fonte da sección: Véxase tamén"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&veaction=edit&section=9" title="Editar a sección: «Bibliografía»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&action=edit&section=9" title="Editar o código fonte da sección: Bibliografía"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A. Kriegl y P. W. Michor: <a rel="nofollow" class="external text" href="http://www.mat.univie.ac.at/~michor/aspects.pdf">"Aspects of the theory of inifinite dimensional manifolds"</a>, <i>Differential Geometry and its Applications</i>, <b>1</b>, 1991, pp. 159–176.</li> <li><a href="/w/index.php?title=Leonida_Tonelli&action=edit&redlink=1" class="new" title="Leonida Tonelli (a páxina aínda non existe)">Leonida Tonelli</a>: <a rel="nofollow" class="external text" href="http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath&idno=ACQ6956">Fondamenti di calcolo delle variazioni</a>, N. Zanichelli, 1921-23</li> <li>Todhunter, I. <a rel="nofollow" class="external text" href="http://www.archive.org/details/histroyofthecalc033379mbp">A history of the calculus of variations</a>, Chelsea, 1861</li> <li>Carll, L. B. <a rel="nofollow" class="external text" href="http://www.archive.org/details/treatiseonthecal032865mbp">A Treatise On The Calculus Of Variations</a> John Wiley & sons, 1881</li> <li>Hancock, H. <a rel="nofollow" class="external text" href="http://www.archive.org/details/151181775">Lectures on the calculus of variations (the Weierstrassian theory)</a> Cincinnati University Press, 1904</li> <li>Bolza, O <a rel="nofollow" class="external text" href="http://name.umdl.umich.edu/ACM2513.0001.001">Lectures on the calculus of variations</a>, Chicago University Press, 1904</li> <li>Byerly, W. E. <a rel="nofollow" class="external text" href="http://name.umdl.umich.edu/ACQ6938.0001.001">Introduction to the calculus of variations</a> <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>, 1917</li> <li>Weinstock, R. <a rel="nofollow" class="external text" href="http://www.archive.org/details/calculusofvariat033563mbp">Calculus Of Variations With Applications To Physics And Engineering</a>, McGrawHill, 1952</li> <li>Hadamard J. e Fréchet, M. <a rel="nofollow" class="external text" href="http://www.archive.org/details/leconssurlecalcu00hadarich">Leçons sur le calcul des variations</a> (francese) Hermann, 1910</li> <li>Fomin, S.V. and Gelfand, I.M.: Calculus of Variations, Dover Publ., 2000</li> <li>Lebedev, L.P. and Cloud, M.J.: The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics, World Scientific, 2003, pages 1 – 98</li> <li>Charles Fox: An Introduction to the Calculus of Variations, Dover Publ., 1987</li> <li><a href="/w/index.php?title=Giuseppe_Buttazzo&action=edit&redlink=1" class="new" title="Giuseppe Buttazzo (a páxina aínda non existe)">Giuseppe Buttazzo</a>, <a href="/w/index.php?title=Gianni_Dal_Maso&action=edit&redlink=1" class="new" title="Gianni Dal Maso (a páxina aínda non existe)">Gianni Dal Maso</a>, <a href="/w/index.php?title=Ennio_De_Giorgi&action=edit&redlink=1" class="new" title="Ennio De Giorgi (a páxina aínda non existe)">Ennio De Giorgi</a>. <a rel="nofollow" class="external text" href="http://www.treccani.it/enciclopedia/calcolo-delle-variazioni_(8Enciclopedia-Novecento)/">Variazioni, calcolo delle</a>, <i>Enciclopedia del Novecento</i>, II Supplemento (1998), <a href="/w/index.php?title=Istituto_dell%27Enciclopedia_italiana_Treccani&action=edit&redlink=1" class="new" title="Istituto dell'Enciclopedia italiana Treccani (a páxina aínda non existe)">Istituto dell'Enciclopedia italiana Treccani</a></li> <li><a href="/w/index.php?title=Gianni_Dal_Maso&action=edit&redlink=1" class="new" title="Gianni Dal Maso (a páxina aínda non existe)">Gianni Dal Maso</a>, <a rel="nofollow" class="external text" href="http://www.treccani.it/enciclopedia/calcolo-delle-variazioni_%28Enciclopedia-della-Scienza-e-della-Tecnica%29/">Variazioni, calcolo delle</a>, <i>Enciclopedia della Scienza e della Tecnica</i>, (2007), <a href="/w/index.php?title=Istituto_dell%27Enciclopedia_italiana_Treccani&action=edit&redlink=1" class="new" title="Istituto dell'Enciclopedia italiana Treccani (a páxina aínda non existe)">Istituto dell'Enciclopedia italiana Treccani</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Outros_artigos">Outros artigos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&veaction=edit&section=10" title="Editar a sección: «Outros artigos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&action=edit&section=10" title="Editar o código fonte da sección: Outros artigos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Charles-Augustin de Coulomb</a></li> <li><a href="/w/index.php?title=Ecuaci%C3%B3ns_de_Euler-Lagrange&action=edit&redlink=1" class="new" title="Ecuacións de Euler-Lagrange (a páxina aínda non existe)">Ecuacións de Euler-Lagrange</a></li> <li><a href="/w/index.php?title=Derivada_funcional&action=edit&redlink=1" class="new" title="Derivada funcional (a páxina aínda non existe)">Derivada funcional</a></li> <li><a href="/w/index.php?title=Mec%C3%A1nica_de_chans&action=edit&redlink=1" class="new" title="Mecánica de chans (a páxina aínda non existe)">Mecánica de chans</a></li> <li><a href="/w/index.php?title=Teor%C3%ADa_de_Mohr-Coulomb&action=edit&redlink=1" class="new" title="Teoría de Mohr-Coulomb (a páxina aínda non existe)">Teoría de Mohr-Coulomb</a></li> <li><a href="/w/index.php?title=Torsi%C3%B3n_mec%C3%A1nica&action=edit&redlink=1" class="new" title="Torsión mecánica (a páxina aínda non existe)">Torsión mecánica</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Ligazóns_externas"><span id="Ligaz.C3.B3ns_externas"></span>Ligazóns externas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&veaction=edit&section=11" title="Editar a sección: «Ligazóns externas»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%C3%A1lculo_de_variaci%C3%B3ns&action=edit&section=11" title="Editar o código fonte da sección: Ligazóns externas"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Variational_calculus">Variational calculus</a>. <i><a href="/w/index.php?title=Encyclopedia_of_Mathematics&action=edit&redlink=1" class="new" title="Encyclopedia of Mathematics (a páxina aínda non existe)">Encyclopedia of Mathematics</a></i>.</li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/calculusofvariations">calculus of variations</a>. <i><a href="/w/index.php?title=PlanetMath&action=edit&redlink=1" class="new" title="PlanetMath (a páxina aínda non existe)">PlanetMath</a></i>.</li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/CalculusofVariations.html">Calculus of Variations</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170609215523/http://www.exampleproblems.com/wiki/index.php/Calculus_of_Variations">Calculus of variations</a>. Example problems.</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/playlist?list=PL521C2DFD15FF568C">Mathematics - Calculus of Variations and Integral Equations</a>. Lectures on <a href="/wiki/YouTube" title="YouTube">YouTube</a>.</li> <li>Selected papers on Geodesic Fields. <a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/geodesic_fields_-_pt._1.pdf">Part I</a>, <a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/geodesic_fields_-_pt._2.pdf">Part II</a>.</li></ul> <div role="navigation" class="navbox" aria-labelledby="Control_de_autoridades" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Control_de_autoridades" scope="row" class="navbox-group" style="width:1%;width: 12%; text-align:center;"><a href="/wiki/Axuda:Control_de_autoridades" title="Axuda:Control de autoridades">Control de autoridades</a></th><td class="navbox-list navbox-odd plainlinks" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div 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