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</div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Variationsrechnung</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">aus Wikipedia, der freien Enzyklopädie</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a> <a class="mw-jump-link" href="#searchInput">Zur Suche springen</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Variationsrechnung</b> ist ein <a href="/wiki/Mathematik" title="Mathematik">mathematisches</a> Teilgebiet der <a href="/wiki/Analysis" title="Analysis">Analysis</a>, in welchem kleine Änderungen in <a href="/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a> und <a href="/wiki/Funktional" title="Funktional">Funktionalen</a> studiert werden, um Minima und Maxima von Funktionalen zu bestimmen. </p><p>Dieses (unendlichdimensionale) <a href="/wiki/Optimierungsproblem" title="Optimierungsproblem">Optimierungsproblem</a> mit Anwendungen in der <a href="/wiki/Theoretische_Physik" class="mw-redirect" title="Theoretische Physik">theoretischen</a> und der <a href="/wiki/Mathematische_Physik" title="Mathematische Physik">mathematischen Physik</a> wurde um die Mitte des 18. Jahrhunderts insbesondere von <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> und <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> zu einem Fachgebiet entwickelt.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Die Variationsrechnung, ihre verwandten Themen und Anwendungen sind Gegenstand aktueller Lehre,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Weiterentwicklung<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> und Forschung.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Die Frage <i>„Wie können die Methoden der Variationsrechnung weiterentwickelt werden?“</i> ist das 23. Problem auf <a href="/wiki/Hilbertsche_Probleme" title="Hilbertsche Probleme">Hilberts Liste</a>. Weitere Beiträge lieferten u. a. die Mathematiker <a href="/wiki/Ennio_De_Giorgi" title="Ennio De Giorgi">Ennio De Giorgi</a> und <a href="/wiki/Charles_Morrey" title="Charles Morrey">Charles Morrey</a>. Ihre Forschungsarbeiten führte zur Lösung des 19. Hilbert-Problems mit der Herausforderung <i>„Sind alle Lösungen von regulären Variationsproblemen analytisch?“.</i> Die von der deutschen Mathematikerin <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a> entwickelten <a href="/wiki/Noether-Theorem" title="Noether-Theorem">Theoreme</a>, die mit der Variationsrechnung zusammenhängen,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> spielen heutzutage eine bedeutende Rolle in der modernen Physik (<a href="/wiki/Symmetrie_(Physik)" title="Symmetrie (Physik)">Symmetrie</a>). Der US-Mathematikerin <a href="/wiki/Karen_Uhlenbeck" title="Karen Uhlenbeck">Karen Uhlenbeck</a> wurde 2019 der <a href="/wiki/Abelpreis" title="Abelpreis">Abelpreis</a> zugesprochen.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Uhlenbeck hat sich intensiv mit der Variationsrechnung befasst.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Grundlagen"><span class="tocnumber">1</span> <span class="toctext">Grundlagen</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Anwendungsgebiete"><span class="tocnumber">2</span> <span class="toctext">Anwendungsgebiete</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Historisch"><span class="tocnumber">2.1</span> <span class="toctext">Historisch</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Ingenieurwesen"><span class="tocnumber">2.2</span> <span class="toctext">Ingenieurwesen</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Mathematik"><span class="tocnumber">2.3</span> <span class="toctext">Mathematik</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Physik"><span class="tocnumber">2.4</span> <span class="toctext">Physik</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-7"><a href="#Direkte_Methoden_der_Variationsrechnung"><span class="tocnumber">3</span> <span class="toctext">Direkte Methoden der Variationsrechnung</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Euler-Lagrange-Gleichung"><span class="tocnumber">4</span> <span class="toctext">Euler-Lagrange-Gleichung</span></a> <ul> <li class="toclevel-2 tocsection-9"><a href="#Ein_Hilfsmittel_aus_der_Analysis_reeller_Funktionen_in_einer_reellen_Veränderlichen"><span class="tocnumber">4.1</span> <span class="toctext">Ein Hilfsmittel aus der Analysis reeller Funktionen in einer reellen Veränderlichen</span></a></li> <li class="toclevel-2 tocsection-10"><a href="#Euler-Lagrange-Gleichung;_Variationsableitung;_weitere_notwendige_bzw._hinreichende_Bedingungen"><span class="tocnumber">4.2</span> <span class="toctext">Euler-Lagrange-Gleichung; Variationsableitung; weitere notwendige bzw. hinreichende Bedingungen</span></a> <ul> <li class="toclevel-3 tocsection-11"><a href="#Bemerkungen"><span class="tocnumber">4.2.1</span> <span class="toctext">Bemerkungen</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-12"><a href="#Verallgemeinerung_für_höhere_Ableitung_und_Dimensionen"><span class="tocnumber">4.3</span> <span class="toctext">Verallgemeinerung für höhere Ableitung und Dimensionen</span></a></li> <li class="toclevel-2 tocsection-13"><a href="#Weiterführende_Verallgemeinerungen"><span class="tocnumber">4.4</span> <span class="toctext">Weiterführende Verallgemeinerungen</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-14"><a href="#Siehe_auch"><span class="tocnumber">5</span> <span class="toctext">Siehe auch</span></a></li> <li class="toclevel-1 tocsection-15"><a href="#Weblinks"><span class="tocnumber">6</span> <span class="toctext">Weblinks</span></a> <ul> <li class="toclevel-2 tocsection-16"><a href="#Journale_&_andere_Beiträge"><span class="tocnumber">6.1</span> <span class="toctext">Journale & andere Beiträge</span></a></li> <li class="toclevel-2 tocsection-17"><a href="#Schools_&_Workshops"><span class="tocnumber">6.2</span> <span class="toctext">Schools & Workshops</span></a></li> <li class="toclevel-2 tocsection-18"><a href="#Skripte"><span class="tocnumber">6.3</span> <span class="toctext">Skripte</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-19"><a href="#Literatur"><span class="tocnumber">7</span> <span class="toctext">Literatur</span></a> <ul> <li class="toclevel-2 tocsection-20"><a href="#Moderne_Lehrbücher"><span class="tocnumber">7.1</span> <span class="toctext">Moderne Lehrbücher</span></a></li> <li class="toclevel-2 tocsection-21"><a href="#Monografien"><span class="tocnumber">7.2</span> <span class="toctext">Monografien</span></a></li> <li class="toclevel-2 tocsection-22"><a href="#Klassische_und_historische_Werke"><span class="tocnumber">7.3</span> <span class="toctext">Klassische und historische Werke</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-23"><a href="#Einzelnachweise"><span class="tocnumber">8</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Grundlagen"><span id="Station.C3.A4re_Funktion"></span><span id="Stationäre_Funktion"></span> Grundlagen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=1" title="Abschnitt bearbeiten: Grundlagen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Grundlagen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Variationsrechnung beschäftigt sich mit reellen <a href="/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a> von Funktionen, die auch <a href="/wiki/Funktional" title="Funktional">Funktionale</a> genannt werden. Solche Funktionale können etwa Integrale über eine unbekannte Funktion und ihre Ableitungen sein. Dabei interessiert man sich für <b>stationäre Funktionen</b>, also solche, für die das Funktional ein <a href="/wiki/Extremwert" title="Extremwert">Maximum</a>, ein <a href="/wiki/Extremwert" title="Extremwert">Minimum</a> (Extremale)<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> oder einen <a href="/wiki/Sattelpunkt" title="Sattelpunkt">Sattelpunkt</a> annimmt. Einige klassische Probleme können elegant mit Hilfe von Funktionalen formuliert werden. </p><p>Das Schlüssel<a href="/wiki/Theorem" title="Theorem">theorem</a> der Variationsrechnung ist die <b>Euler-Lagrange-Gleichung</b>, genauer „Euler-Lagrange’sche Differentialgleichung“. Diese beschreibt die Stationaritätsbedingung eines Funktionals. Wie bei der Aufgabe, die Maxima und Minima einer Funktion zu bestimmen, wird sie aus der Analyse kleiner Änderungen (Variation) um die angenommene Lösung hergeleitet. Die Euler-Lagrangesche Differentialgleichung ist lediglich eine <a href="/wiki/Notwendige_und_hinreichende_Bedingung" title="Notwendige und hinreichende Bedingung">notwendige Bedingung</a>. Weitere notwendige Bedingungen für das Vorliegen einer Extremalen lieferten <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a> und <a href="/wiki/Alfred_Clebsch" title="Alfred Clebsch">Alfred Clebsch</a> sowie <a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Carl Gustav Jacob Jacobi</a>. Eine hinreichende, aber nicht notwendige Bedingung stammt von <a href="/wiki/Karl_Weierstra%C3%9F" title="Karl Weierstraß">Karl Weierstraß</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Weierstraß präsentierte ein Gegenbeispiel zum <a href="/wiki/Dirichlet-Prinzip" title="Dirichlet-Prinzip">Dirichletschen Prinzip</a>. Basierend auf dieser neuen Erkenntnis („Existenztheorien“)<sup id="cite_ref-:0_12-0" class="reference"><a href="#cite_note-:0-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> entwickelte sich die Variationsrechnung fortan zu den <b>Direkten Methoden der Variationsrechnung.</b><sup id="cite_ref-:0_12-1" class="reference"><a href="#cite_note-:0-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Anwendungsgebiete">Anwendungsgebiete</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=2" title="Abschnitt bearbeiten: Anwendungsgebiete" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Anwendungsgebiete"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Historisch">Historisch</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=3" title="Abschnitt bearbeiten: Historisch" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Historisch"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ein typisches Anwendungsbeispiel ist das <a href="/wiki/Brachistochrone" title="Brachistochrone">Brachistochronenproblem</a>: Auf welcher Kurve in einem Schwerefeld von einem Punkt A zu einem Punkt B, der unterhalb, aber nicht direkt unter A liegt, benötigt ein Objekt die geringste Zeit zum Durchlaufen der Kurve? Von allen Kurven zwischen A und B minimiert eine den Ausdruck, der die Zeit des Durchlaufens der Kurve beschreibt. Dieser Ausdruck ist ein Integral, das die unbekannte, gesuchte Funktion, die die Kurve von A nach B beschreibt, und deren Ableitungen enthält. </p> <div class="mw-heading mw-heading3"><h3 id="Ingenieurwesen">Ingenieurwesen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=4" title="Abschnitt bearbeiten: Ingenieurwesen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Ingenieurwesen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Variationsrechnung findet Anwendung in der Steuerungs- und Regelungstheorie, wenn es um die Bestimmung von <a href="/wiki/Optimale_Regelung" title="Optimale Regelung">Optimalreglern</a> geht. Die aus dem <a href="/wiki/Rayleigh-Ritz-Prinzip" title="Rayleigh-Ritz-Prinzip">Verfahren von Ritz</a> weiterentwickelte <a href="/wiki/Finite-Elemente-Methode" title="Finite-Elemente-Methode">Finite-Elemente-Methode</a> findet z. B. Anwendung in der <a href="/wiki/Strukturmechanik" title="Strukturmechanik">Strukturmechanik</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Mathematik">Mathematik</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=5" title="Abschnitt bearbeiten: Mathematik" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Mathematik"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Methoden der Variationsrechnung tauchen bei den <a href="/wiki/Hilbertraum" title="Hilbertraum">Hilbertraum</a>-Techniken, der <a href="/wiki/Morsetheorie" class="mw-redirect" title="Morsetheorie">Morsetheorie</a> und bei der <a href="/wiki/Symplektische_Mannigfaltigkeit" title="Symplektische Mannigfaltigkeit">symplektischen Geometrie</a> auf. Der Begriff <i>Variation</i> wird für alle „Extremal-Probleme“ von Funktionen verwendet. <a href="/wiki/Geod%C3%A4sie" title="Geodäsie">Geodäsie</a> und <a href="/wiki/Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a> sind Bereiche der Mathematik, in denen Variationen eine Rolle spielen, bzw. mittels dieser weiterentwickelt wird.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Besonders am Problem der <i><a href="/wiki/Minimalfl%C3%A4che" title="Minimalfläche">minimalen Oberflächen</a> (vgl. auch <a href="/wiki/Plateau-Problem" title="Plateau-Problem">Plateau-Problem</a>)</i>, die etwa bei Seifenblasen auftreten, wurde viel gearbeitet.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Variationsmethoden finden Anwendung bei den <a href="/wiki/Partielle_Differentialgleichung" title="Partielle Differentialgleichung">partiellen Differentialgleichungen</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In der Mathematik wurde die Variationsrechnung beispielsweise bei der <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">riemannschen</a> Behandlung des <i>Dirichlet-Prinzips</i> für <a href="/wiki/Harmonische_Funktion" title="Harmonische Funktion">harmonische Funktionen</a> verwendet.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Andere Weiterentwicklungen existieren, z. B. <a href="/wiki/%CE%93-Konvergenz" title="Γ-Konvergenz">Γ-Konvergenz</a>, <a href="/w/index.php?title=Stochastische_Variationsmethoden&action=edit&redlink=1" class="new" title="Stochastische Variationsmethoden (Seite nicht vorhanden)">stochastische Variationsmethoden</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Physik">Physik</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=6" title="Abschnitt bearbeiten: Physik" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Physik"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Variationsrechnung ist die mathematische Grundlage aller physikalischen Extremalprinzipien und deshalb besonders in der <a href="/wiki/Theoretische_Physik" class="mw-redirect" title="Theoretische Physik">theoretischen Physik</a> wichtig, so etwa im <a href="/wiki/Lagrange-Formalismus" title="Lagrange-Formalismus">Lagrange-Formalismus</a> der <a href="/wiki/Klassische_Mechanik" title="Klassische Mechanik">klassischen Mechanik</a> bzw. der <a href="/wiki/Bahnbestimmung" title="Bahnbestimmung">Bahnbestimmung</a>, in der <a href="/wiki/Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> in Anwendung des <a href="/wiki/Hamiltonsches_Prinzip" title="Hamiltonsches Prinzip">Prinzips der kleinsten Wirkung</a> und in der <a href="/wiki/Statistische_Physik" title="Statistische Physik">statistischen Physik</a> im Rahmen der <a href="/wiki/Dichtefunktionaltheorie_(statistische_Physik)" title="Dichtefunktionaltheorie (statistische Physik)">Dichtefunktionaltheorie</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Direkte_Methoden_der_Variationsrechnung">Direkte Methoden der Variationsrechnung</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=7" title="Abschnitt bearbeiten: Direkte Methoden der Variationsrechnung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Direkte Methoden der Variationsrechnung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grundlegend ist der Euler-Lagrange Ansatz effektiv im Auffinden von Extremalen von Funktionalen. Jedoch ist bereits bei mehreren Variablen, wo die Euler-Lagrange-Gleichung eine partielle Differentialgleichung darstellt, es nicht möglich eine explizite Lösung zu finden. Weitere Einschränkungen existieren.<sup id="cite_ref-:0_12-2" class="reference"><a href="#cite_note-:0-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Hingegen gehen die sog. <i>Direkten Methoden der Variationsrechnung</i> auf den generellen Fall ein, welcher der Frage nachgeht, <i>unter welchen generellen Bedingungen können Funktionale minimiert werden?</i><sup id="cite_ref-:0_12-3" class="reference"><a href="#cite_note-:0-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Die Ansätze bedienen sich dabei stark aus den Methoden der <a href="/wiki/Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Wichtige Theoreme und Beiträge zur Methode erfolgten durch <a href="/wiki/Leonida_Tonelli" title="Leonida Tonelli">Tonelli</a>, <a href="/wiki/Guido_Ascoli" title="Guido Ascoli">Ascoli</a>, <a href="/wiki/Cesare_Arzel%C3%A0" title="Cesare Arzelà">Arzelà</a> und <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Zu den direkten Methoden der Variationsrechnung zählen z. B. das <a href="/wiki/Rayleigh-Ritz-Prinzip" title="Rayleigh-Ritz-Prinzip">Approximationsverfahren nach Ritz</a> oder das <a href="/wiki/Differenzenverfahren" class="mw-redirect" title="Differenzenverfahren">Differenzenverfahren</a> nach Euler.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Die mathematische Theorie dazu hat Zusammenhänge mit der Theorie der <a href="/wiki/Konvexe_und_konkave_Funktionen" title="Konvexe und konkave Funktionen">Konvexen Analysis</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Euler-Lagrange-Gleichung">Euler-Lagrange-Gleichung</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=8" title="Abschnitt bearbeiten: Euler-Lagrange-Gleichung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Euler-Lagrange-Gleichung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zentrales Element bildet die <b>Euler-Lagrange-Gleichung</b><sup id="cite_ref-:0_12-4" class="reference"><a href="#cite_note-:0-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta I(x,\delta x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>δ<!-- δ --></mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta I(x,\delta x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d2a7ed20b3a43bf8c60db88bd391d32532e1d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.033ex; height:2.843ex;" alt="{\displaystyle \delta I(x,\delta x)=0}"></span>,</dd></dl> <p>die für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle I=\int {\mathcal {L}}\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle I=\int {\mathcal {L}}\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d19ddedfa6b283250e504de16d0b77eb6ab4d79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.198ex; height:3.176ex;" alt="{\textstyle I=\int {\mathcal {L}}\,\mathrm {d} t}"></span> gerade zur <a href="/wiki/Lagrange-Gleichung" class="mw-redirect" title="Lagrange-Gleichung">Lagrange-Gleichung</a> aus der <a href="/wiki/Klassische_Mechanik" title="Klassische Mechanik">klassischen Mechanik</a> wird. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> ist dabei die Variation, <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 {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span> die sog. Lagrangefunktion und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> das Funktional. Mehr dazu siehe die Herleitung. </p> <div class="mw-heading mw-heading3"><h3 id="Ein_Hilfsmittel_aus_der_Analysis_reeller_Funktionen_in_einer_reellen_Veränderlichen"><span id="Ein_Hilfsmittel_aus_der_Analysis_reeller_Funktionen_in_einer_reellen_Ver.C3.A4nderlichen"></span>Ein Hilfsmittel aus der Analysis reeller Funktionen in einer reellen Veränderlichen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=9" title="Abschnitt bearbeiten: Ein Hilfsmittel aus der Analysis reeller Funktionen in einer reellen Veränderlichen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Ein Hilfsmittel aus der Analysis reeller Funktionen in einer reellen Veränderlichen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Im Folgenden wird eine wichtige Technik der Variationsrechnung demonstriert, bei der eine notwendige Aussage für eine lokale Minimumstelle einer <a href="/wiki/Reelle_Funktion" class="mw-redirect" title="Reelle Funktion">reellen Funktion</a> mit nur einer reellen Veränderlichen in eine notwendige Aussage für eine lokale Minimumstelle eines Funktionals übertragen wird. Diese Aussage kann dann oftmals zum Aufstellen beschreibender Gleichungen für stationäre Funktionen eines Funktionals benutzt werden. </p><p>Sei ein <a href="/wiki/Funktional" title="Funktional">Funktional</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 I\colon X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>:<!-- : --></mo> <mi>X</mi> <mo stretchy="false">→<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\colon X\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad820d04b96eab857b71b3eacce4ffe3a61c68e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.478ex; height:2.176ex;" alt="{\displaystyle I\colon X\to \mathbb {R} }"></span> auf einem Funktionenraum <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> gegeben (<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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> muss mind. ein <a href="/wiki/Topologischer_Raum" title="Topologischer Raum">topologischer Raum</a> sein). Das Funktional habe an der Stelle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈<!-- ∈ --></mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span> ein <a href="/wiki/Extremwert" title="Extremwert">lokales Minimum</a>. </p><p>Durch den folgenden einfachen Trick tritt an die Stelle des „schwierig handhabbaren“ Funktionals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> eine reelle Funktion <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(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30d1603b950a4f18706aeb41b4c1e1c256026c5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle F(\alpha )}"></span>, die nur von einem reellen Parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> abhängt „und entsprechend einfacher zu behandeln ist“. </p><p>Mit einem <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 \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ<!-- ϵ --></mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span> sei <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_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>∈<!-- ∈ --></mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>ϵ<!-- ϵ --></mi> <mo>,</mo> <mi>ϵ<!-- ϵ --></mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500983860ea85f72043d1716a91dc513eb4b92e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.154ex; height:3.176ex;" alt="{\displaystyle (x_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}}"></span> eine beliebige stetig durch den reellen Parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> parametrisierte Familie von Funktionen <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_{\alpha }\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo>∈<!-- ∈ --></mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\alpha }\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2de3f1eebb49760b3bd7cd30be086592ee53ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.435ex; height:2.509ex;" alt="{\displaystyle x_{\alpha }\in X}"></span>. Dabei sei die Funktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> (d. h., <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_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd68505f66718efda85db1b17010ffa0222054f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.614ex; height:2.009ex;" alt="{\displaystyle x_{\alpha }}"></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cc00f65bbc630448311dd2dc82e7ce5e90985a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =0}"></span>) gerade gleich der stationären Funktion <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>. Außerdem sei die durch die Gleichung </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\alpha ):=I(x_{\alpha })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\alpha ):=I(x_{\alpha })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd18238495729c687ad3fc2156d4d144b16b5fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.378ex; height:2.843ex;" alt="{\displaystyle F(\alpha ):=I(x_{\alpha })}"></span></dd></dl> <p>definierte Funktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon (-\epsilon ,\epsilon )\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:<!-- : --></mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>ϵ<!-- ϵ --></mi> <mo>,</mo> <mi>ϵ<!-- ϵ --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\colon (-\epsilon ,\epsilon )\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fcb14d39f4667ba4346de5c58974fd7eb66540" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.607ex; height:2.843ex;" alt="{\displaystyle F\colon (-\epsilon ,\epsilon )\to \mathbb {R} }"></span> an der Stelle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cc00f65bbc630448311dd2dc82e7ce5e90985a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =0}"></span> differenzierbar. </p><p>Die stetige Funktion <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> nimmt dann an der Stelle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cc00f65bbc630448311dd2dc82e7ce5e90985a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =0}"></span> ein lokales Minimum an, da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14745b8f7723f2031eb388152d53b1b39b43bb66" 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_{0}=x}"></span> ein lokales Minimum von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> ist. </p><p>Aus der Analysis für reelle Funktionen in einer reellen Veränderlichen ist bekannt, dass dann <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 \textstyle {\frac {\mathrm {d} }{\mathrm {d} \alpha }}F(\alpha )|_{\alpha =0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>α<!-- α --></mi> </mrow> </mfrac> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\mathrm {d} }{\mathrm {d} \alpha }}F(\alpha )|_{\alpha =0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26beff0d9797f972f1264af08a17b961edd9a8f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.132ex; height:3.843ex;" alt="{\displaystyle \textstyle {\frac {\mathrm {d} }{\mathrm {d} \alpha }}F(\alpha )|_{\alpha =0}=0}"></span> gilt. Auf das Funktional übertragen heißt das </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\frac {\mathrm {d} }{\mathrm {d} \alpha }}I(x_{\alpha })\right|_{\alpha =0}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>α<!-- α --></mi> </mrow> </mfrac> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {\mathrm {d} }{\mathrm {d} \alpha }}I(x_{\alpha })\right|_{\alpha =0}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06e79c6675a4c3363555781f1b2917ae52500f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.15ex; height:5.843ex;" alt="{\displaystyle \left.{\frac {\mathrm {d} }{\mathrm {d} \alpha }}I(x_{\alpha })\right|_{\alpha =0}=0.}"></span></dd></dl> <p>Beim Aufstellen der gewünschten Gleichungen für stationäre Funktionen wird dann noch ausgenutzt, dass die vorstehende Gleichung für jede beliebige („gutartige“) Familie <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_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>∈<!-- ∈ --></mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>ϵ<!-- ϵ --></mi> <mo>,</mo> <mi>ϵ<!-- ϵ --></mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500983860ea85f72043d1716a91dc513eb4b92e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.154ex; height:3.176ex;" alt="{\displaystyle (x_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}}"></span> mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14745b8f7723f2031eb388152d53b1b39b43bb66" 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_{0}=x}"></span> gelten muss. </p><p>Das soll im nächsten Abschnitt anhand der Euler-Gleichung demonstriert werden. </p> <div class="mw-heading mw-heading3"><h3 id="Euler-Lagrange-Gleichung;_Variationsableitung;_weitere_notwendige_bzw._hinreichende_Bedingungen"><span id="Euler-Lagrange-Gleichung.3B_Variationsableitung.3B_weitere_notwendige_bzw._hinreichende_Bedingungen"></span>Euler-Lagrange-Gleichung; Variationsableitung; weitere notwendige bzw. hinreichende Bedingungen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=10" title="Abschnitt bearbeiten: Euler-Lagrange-Gleichung; Variationsableitung; weitere notwendige bzw. hinreichende Bedingungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Euler-Lagrange-Gleichung; Variationsableitung; weitere notwendige bzw. hinreichende Bedingungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gegeben seien zwei Zeitpunkte <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{a},t_{e}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{a},t_{e}\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34cd007b66f694b8140c08928c911267d6748877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.332ex; height:2.509ex;" alt="{\displaystyle t_{a},t_{e}\in \mathbb {R} }"></span> mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{e}>t_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>></mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{e}>t_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf02ce36fa468ad5425c59897a0701129f9f44c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.878ex; height:2.343ex;" alt="{\displaystyle t_{e}>t_{a}}"></span> und eine in allen Argumenten zweifach stetig differenzierbare Funktion, die <a href="/wiki/Lagrangefunktion" class="mw-redirect" title="Lagrangefunktion">Lagrangefunktion</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\colon (t_{a},t_{e})\times G\to \mathbb {R} \ ,\quad G\subset \mathbb {R} ^{n}\times \mathbb {R} ^{n}\ ,\quad G{\text{ offen}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>:<!-- : --></mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×<!-- × --></mo> <mi>G</mi> <mo stretchy="false">→<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mi>G</mi> <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>×<!-- × --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> offen</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\colon (t_{a},t_{e})\times G\to \mathbb {R} \ ,\quad G\subset \mathbb {R} ^{n}\times \mathbb {R} ^{n}\ ,\quad G{\text{ offen}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30d7549dae3442f5ba57fbf45b3aa04584dc8a25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.971ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}\colon (t_{a},t_{e})\times G\to \mathbb {R} \ ,\quad G\subset \mathbb {R} ^{n}\times \mathbb {R} ^{n}\ ,\quad G{\text{ offen}}}"></span>.</dd></dl> <p>Beispielsweise ist bei der Lagrangefunktion des freien relativistischen Teilchens mit Masse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> und <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=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3467f9e219a5ea38a30da5c3a02c2c23f61a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=1}"></span> </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 {\mathcal {L}}(t,x,v)=-m{\sqrt {1-v^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−<!-- − --></mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(t,x,v)=-m{\sqrt {1-v^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af34ea9b274a2a69cc99530b5791e6a248f2755d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.233ex; height:3.509ex;" alt="{\displaystyle {\mathcal {L}}(t,x,v)=-m{\sqrt {1-v^{2}}}}"></span></dd></dl> <p>das Gebiet <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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> das kartesische Produkt von <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 \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> und dem Inneren der <a href="/wiki/Einheitskugel" title="Einheitskugel">Einheitskugel</a>. </p><p>Als Funktionenraum <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> wird die Menge aller zweifach stetig differenzierbaren Funktionen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\colon [t_{a},t_{e}]\to \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>:<!-- : --></mo> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">]</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\colon [t_{a},t_{e}]\to \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/461206b20d0ca8b7ea8fbce2fe12cfdc66d85137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.981ex; height:2.843ex;" alt="{\displaystyle x\colon [t_{a},t_{e}]\to \mathbb {R} ^{n}}"></span></dd></dl> <p>gewählt, die zum Anfangszeitpunkt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1f89579c6b54711128de7bad0f0f760e116180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.941ex; height:2.343ex;" alt="{\displaystyle t_{a}}"></span> und zum Endzeitpunkt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4004a41da2e68ea68ed4f605bb28db65c709c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.343ex;" alt="{\displaystyle t_{e}}"></span> die fest vorgegebenen Orte <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_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc83e1ad0eaf733b246521d51f5880901c563a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.432ex; height:2.009ex;" alt="{\displaystyle x_{a}}"></span> bzw. <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_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bffc1d36b6465edf0ff6f9950096c46f85888c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.328ex; height:2.009ex;" alt="{\displaystyle x_{e}}"></span> einnehmen: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t_{a})=x_{a}\ ,\quad x(t_{e})=x_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t_{a})=x_{a}\ ,\quad x(t_{e})=x_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a13d1affc51c46b1e1f45e0bc4e1715bf8b8d761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.951ex; height:2.843ex;" alt="{\displaystyle x(t_{a})=x_{a}\ ,\quad x(t_{e})=x_{e}}"></span></dd></dl> <p>und deren Werte zusammen mit den Werten ihrer Ableitung in <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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> liegen, </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 \forall t\in [t_{a},t_{e}]\colon \left(x(t),{\frac {\mathrm {d} x}{\mathrm {d} t}}(t)\right)\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>t</mi> <mo>∈<!-- ∈ --></mo> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>:<!-- : --></mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>∈<!-- ∈ --></mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall t\in [t_{a},t_{e}]\colon \left(x(t),{\frac {\mathrm {d} x}{\mathrm {d} t}}(t)\right)\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/914dedf83c57715698f8b4ba969d5b0c70067b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.322ex; height:6.176ex;" alt="{\displaystyle \forall t\in [t_{a},t_{e}]\colon \left(x(t),{\frac {\mathrm {d} x}{\mathrm {d} t}}(t)\right)\in G}"></span>.</dd></dl> <p>Mit der Lagrangefunktion <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 {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span> wird nun das Funktional <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\colon X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>:<!-- : --></mo> <mi>X</mi> <mo stretchy="false">→<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\colon X\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad820d04b96eab857b71b3eacce4ffe3a61c68e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.478ex; height:2.176ex;" alt="{\displaystyle I\colon X\to \mathbb {R} }"></span>, die Wirkung, durch </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 I(x):=\int \limits _{t_{a}}^{t_{e}}{\mathcal {L}}(t,x(t),{\dot {x}}(t))\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munderover> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </munderover> <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>t</mi> <mo>,</mo> <mi>x</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>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(x):=\int \limits _{t_{a}}^{t_{e}}{\mathcal {L}}(t,x(t),{\dot {x}}(t))\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83c257cd0ce1a0aa8a9719fe1f47a0090917f073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:27.613ex; height:9.343ex;" alt="{\displaystyle I(x):=\int \limits _{t_{a}}^{t_{e}}{\mathcal {L}}(t,x(t),{\dot {x}}(t))\,\mathrm {d} t}"></span></dd></dl> <p>definiert. Gesucht ist diejenige Funktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈<!-- ∈ --></mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span>, die die Wirkung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> minimiert. </p><p>Entsprechend der im vorhergehenden Abschnitt vorgestellten Technik untersuchen wir dazu alle differenzierbaren einparametrigen Familien <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_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}\subset X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>∈<!-- ∈ --></mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>ϵ<!-- ϵ --></mi> <mo>,</mo> <mi>ϵ<!-- ϵ --></mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⊂<!-- ⊂ --></mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}\subset X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9a294a4fe463a7c73efbe05c3a88701ba121cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.233ex; height:3.176ex;" alt="{\displaystyle (x_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}\subset X}"></span>, die für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cc00f65bbc630448311dd2dc82e7ce5e90985a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =0}"></span> durch die stationäre Funktion <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> des Funktionals gehen (es gilt also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14745b8f7723f2031eb388152d53b1b39b43bb66" 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_{0}=x}"></span>). Genutzt wird die im letzten Abschnitt hergeleitete Gleichung </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 0=\left.{\frac {\mathrm {d} }{\mathrm {d} \alpha }}I(x_{\alpha })\right|_{\alpha =0}=\left[{\frac {\mathrm {d} }{\mathrm {d} \alpha }}\int \limits _{t_{a}}^{t_{e}}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\,\mathrm {d} t\right]_{\alpha =0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>α<!-- α --></mi> </mrow> </mfrac> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>α<!-- α --></mi> </mrow> </mfrac> </mrow> <munderover> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </munderover> <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>t</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\left.{\frac {\mathrm {d} }{\mathrm {d} \alpha }}I(x_{\alpha })\right|_{\alpha =0}=\left[{\frac {\mathrm {d} }{\mathrm {d} \alpha }}\int \limits _{t_{a}}^{t_{e}}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\,\mathrm {d} t\right]_{\alpha =0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76a1459e05ba4a07e771fc6df3a26168f559796c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:53.216ex; height:9.676ex;" alt="{\displaystyle 0=\left.{\frac {\mathrm {d} }{\mathrm {d} \alpha }}I(x_{\alpha })\right|_{\alpha =0}=\left[{\frac {\mathrm {d} }{\mathrm {d} \alpha }}\int \limits _{t_{a}}^{t_{e}}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\,\mathrm {d} t\right]_{\alpha =0}}"></span>.</dd></dl> <p>Hereinziehen der Differentiation nach dem Parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> in das Integral liefert mit der Kettenregel (<a href="/wiki/Leibnizregel_f%C3%BCr_Parameterintegrale" title="Leibnizregel für Parameterintegrale">Leibnizregel für Parameterintegrale</a>) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}0&=\left[\int \limits _{t_{a}}^{t_{e}}\left(\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }x_{\alpha }(t)+\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }{\dot {x}}_{\alpha }(t)\right)\,\mathrm {d} t\right]_{\alpha =0}\\&=\left[\int \limits _{t_{a}}^{t_{e}}\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t+\int \limits _{t_{a}}^{t_{e}}\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }{\dot {x}}_{\alpha }(t)\,\mathrm {d} t\right]_{\alpha =0}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </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> <mi>t</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </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> <mi>t</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </munderover> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </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> <mi>t</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>+</mo> <munderover> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </munderover> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </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> <mi>t</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0&=\left[\int \limits _{t_{a}}^{t_{e}}\left(\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }x_{\alpha }(t)+\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }{\dot {x}}_{\alpha }(t)\right)\,\mathrm {d} t\right]_{\alpha =0}\\&=\left[\int \limits _{t_{a}}^{t_{e}}\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t+\int \limits _{t_{a}}^{t_{e}}\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }{\dot {x}}_{\alpha }(t)\,\mathrm {d} t\right]_{\alpha =0}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a691c6348a16a73e1badea19ed638f7ea8ccbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:79.375ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}0&=\left[\int \limits _{t_{a}}^{t_{e}}\left(\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }x_{\alpha }(t)+\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }{\dot {x}}_{\alpha }(t)\right)\,\mathrm {d} t\right]_{\alpha =0}\\&=\left[\int \limits _{t_{a}}^{t_{e}}\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t+\int \limits _{t_{a}}^{t_{e}}\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }{\dot {x}}_{\alpha }(t)\,\mathrm {d} t\right]_{\alpha =0}.\end{aligned}}}"></span></dd></dl> <p>Dabei stehen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{2},\partial _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{2},\partial _{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7828acbf8bc7259dcb6e6b3b62a0b4df0b2e6290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.611ex; height:2.509ex;" alt="{\displaystyle \partial _{2},\partial _{3}}"></span> für die Ableitungen nach dem zweiten bzw. dritten Argument und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eab9f5f18c7860f682fdf1443b92c30cc7e02e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.519ex; height:2.509ex;" alt="{\displaystyle \partial _{\alpha }}"></span> für die partielle Ableitung nach dem Parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. </p><p>Es wird sich später als günstig erweisen, wenn im zweiten Integral statt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\alpha }{\dot {x}}_{\alpha }(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\alpha }{\dot {x}}_{\alpha }(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ff5934a68df75f9b9ec9c47fb82613789f6120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.781ex; height:2.843ex;" alt="{\displaystyle \partial _{\alpha }{\dot {x}}_{\alpha }(t)}"></span> wie im ersten Integral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\alpha }x_{\alpha }(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\alpha }x_{\alpha }(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75ee178b770d6c7c5073e9af41d61db2212e4722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.781ex; height:2.843ex;" alt="{\displaystyle \partial _{\alpha }x_{\alpha }(t)}"></span> steht. Das erreicht man durch partielle Integration: </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 0=\left.\left.\left[\int \limits _{t_{a}}^{t_{e}}\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\,\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t+\right[\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\,\partial _{\alpha }x_{\alpha }(t)\right]_{t=t_{a}}^{t_{e}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <msubsup> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </munderover> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </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> <mi>t</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>+</mo> </mrow> <mo>[</mo> </mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </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> <mi>t</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </msubsup> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\left.\left.\left[\int \limits _{t_{a}}^{t_{e}}\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\,\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t+\right[\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\,\partial _{\alpha }x_{\alpha }(t)\right]_{t=t_{a}}^{t_{e}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfff00226c8098c5945a06a4ff970efdcfb7412f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:74.561ex; height:10.176ex;" alt="{\displaystyle 0=\left.\left.\left[\int \limits _{t_{a}}^{t_{e}}\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\,\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t+\right[\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\,\partial _{\alpha }x_{\alpha }(t)\right]_{t=t_{a}}^{t_{e}}\right.}"></span> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\left.\int \limits _{t_{a}}^{t_{e}}{\frac {\mathrm {d} }{\mathrm {d} t}}\left(\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\right)\,\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t\right]_{\alpha =0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <munderover> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </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> <mi>t</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\left.\int \limits _{t_{a}}^{t_{e}}{\frac {\mathrm {d} }{\mathrm {d} t}}\left(\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\right)\,\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t\right]_{\alpha =0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2946be34d55c4fce66f72cf34a9b8e36a2ed3c7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:44.877ex; height:9.676ex;" alt="{\displaystyle -\left.\int \limits _{t_{a}}^{t_{e}}{\frac {\mathrm {d} }{\mathrm {d} t}}\left(\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\right)\,\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t\right]_{\alpha =0}}"></span></dd></dl></dd></dl> <p>An den Stellen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b83ca5cb9daf325bc13bf9a0dc87e0de01a5b69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.88ex; height:2.343ex;" alt="{\displaystyle t=t_{a}}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e53a37f8ce94827ecb14d5a7d7a18436898dc66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.776ex; height:2.343ex;" alt="{\displaystyle t=t_{e}}"></span> gelten unabhängig von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> die Bedingungen <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_{\alpha }(t_{a})=x_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\alpha }(t_{a})=x_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f24a5c22190195b27b0ce77078b24440c4a9650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.895ex; height:2.843ex;" alt="{\displaystyle x_{\alpha }(t_{a})=x_{a}}"></span> und <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_{\alpha }(t_{e})=x_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\alpha }(t_{e})=x_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c7580afa76e8d1369bd6398e8afbcd038e361b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.688ex; height:2.843ex;" alt="{\displaystyle x_{\alpha }(t_{e})=x_{e}}"></span>. Ableiten dieser beiden Konstanten nach <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> liefert <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\alpha }x_{\alpha }(t_{a})=\partial _{\alpha }x_{\alpha }(t_{e})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\alpha }x_{\alpha }(t_{a})=\partial _{\alpha }x_{\alpha }(t_{e})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4edaf7f346c3c99bf923e36b6b7800e744dc2af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.022ex; height:2.843ex;" alt="{\displaystyle \partial _{\alpha }x_{\alpha }(t_{a})=\partial _{\alpha }x_{\alpha }(t_{e})=0}"></span>. Deshalb verschwindet der Term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }x_{\alpha }(t)\right]_{t=t_{a}}^{t_{e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>[</mo> <mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </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> <mi>t</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }x_{\alpha }(t)\right]_{t=t_{a}}^{t_{e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9a10af11e272a928bc299525748d4776399657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:31.778ex; height:3.509ex;" alt="{\displaystyle \left[\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\partial _{\alpha }x_{\alpha }(t)\right]_{t=t_{a}}^{t_{e}}}"></span> und man erhält nach Zusammenfassen der Integrale und Ausklammern von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\alpha }x_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\alpha }x_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db39098fe6708d98862a98ef9e5c83009e624dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.132ex; height:2.509ex;" alt="{\displaystyle \partial _{\alpha }x_{\alpha }}"></span> die Gleichung </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 0=\left[\int \limits _{t_{a}}^{t_{e}}\left(\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\right)\,\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t\right]_{\alpha =0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <msub> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </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> <mi>t</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </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> <mi>t</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\left[\int \limits _{t_{a}}^{t_{e}}\left(\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\right)\,\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t\right]_{\alpha =0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/832a40eb56ee84587b72bcf4caa4ed5d260e7fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:72.08ex; height:9.676ex;" alt="{\displaystyle 0=\left[\int \limits _{t_{a}}^{t_{e}}\left(\partial _{2}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}(t,x_{\alpha }(t),{\dot {x}}_{\alpha }(t))\right)\,\partial _{\alpha }x_{\alpha }(t)\,\mathrm {d} t\right]_{\alpha =0}}"></span></dd></dl> <p>und mit <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_{\alpha }(t)|_{\alpha =0}=x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\alpha }(t)|_{\alpha =0}=x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bd1f6f24ed043c6350a94fa227bba89da6298b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.371ex; height:3.009ex;" alt="{\displaystyle x_{\alpha }(t)|_{\alpha =0}=x(t)}"></span> </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 0=\int \limits _{t_{a}}^{t_{e}}\left(\partial _{2}{\mathcal {L}}\left(t,x\left(t\right),{\dot {x}}\left(t\right)\right)-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}\left(t,x\left(t\right),{\dot {x}}\left(t\right)\right)\right)\left[\partial _{\alpha }x_{\alpha }(t)\right]_{\alpha =0}\,\mathrm {d} t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <munderover> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mrow> <mo>[</mo> <mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\int \limits _{t_{a}}^{t_{e}}\left(\partial _{2}{\mathcal {L}}\left(t,x\left(t\right),{\dot {x}}\left(t\right)\right)-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}\left(t,x\left(t\right),{\dot {x}}\left(t\right)\right)\right)\left[\partial _{\alpha }x_{\alpha }(t)\right]_{\alpha =0}\,\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74cf5775bd3638bf340d210a5625f43f8bc8c978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:68.493ex; height:9.343ex;" alt="{\displaystyle 0=\int \limits _{t_{a}}^{t_{e}}\left(\partial _{2}{\mathcal {L}}\left(t,x\left(t\right),{\dot {x}}\left(t\right)\right)-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}\left(t,x\left(t\right),{\dot {x}}\left(t\right)\right)\right)\left[\partial _{\alpha }x_{\alpha }(t)\right]_{\alpha =0}\,\mathrm {d} t.}"></span></dd></dl> <p>Außer zum Anfangszeitpunkt und zum Endzeitpunkt unterliegt <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_{\alpha }(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\alpha }(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c9a07848ba4da3e2bfec20ba0202bb908c82302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.263ex; height:2.843ex;" alt="{\displaystyle x_{\alpha }(t)}"></span> keinen Einschränkungen. Damit sind die Zeitfunktionen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto \left[\partial _{\alpha }x_{\alpha }(t)\right]_{\alpha =0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">↦<!-- ↦ --></mo> <msub> <mrow> <mo>[</mo> <mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\mapsto \left[\partial _{\alpha }x_{\alpha }(t)\right]_{\alpha =0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256a743e9eb95acc689c8d784aa1394d54571f30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.913ex; height:3.009ex;" alt="{\displaystyle t\mapsto \left[\partial _{\alpha }x_{\alpha }(t)\right]_{\alpha =0}}"></span> bis auf die Bedingungen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\alpha }x_{\alpha }(t_{a})=\partial _{\alpha }x_{\alpha }(t_{e})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\alpha }x_{\alpha }(t_{a})=\partial _{\alpha }x_{\alpha }(t_{e})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4edaf7f346c3c99bf923e36b6b7800e744dc2af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.022ex; height:2.843ex;" alt="{\displaystyle \partial _{\alpha }x_{\alpha }(t_{a})=\partial _{\alpha }x_{\alpha }(t_{e})=0}"></span> beliebige zweimal stetig differenzierbare Zeitfunktionen. Die letzte Gleichung kann nach dem <a href="/wiki/Fundamentallemma_der_Variationsrechnung" title="Fundamentallemma der Variationsrechnung">Fundamentallemma der Variationsrechnung</a> also nur dann für alle zulässigen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\partial _{\alpha }x_{\alpha }\right]_{\alpha =0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>[</mo> <mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[\partial _{\alpha }x_{\alpha }\right]_{\alpha =0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b1e0f40d8652eed2e80a387612bba89e37e9d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.811ex; height:3.009ex;" alt="{\displaystyle \left[\partial _{\alpha }x_{\alpha }\right]_{\alpha =0}}"></span> erfüllt sein, wenn der Faktor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \partial _{2}{\mathcal {L}}(t,x(t),{\dot {x}}(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}(t,x(t),{\dot {x}}(t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </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> <mi>t</mi> <mo>,</mo> <mi>x</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>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </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> <mi>t</mi> <mo>,</mo> <mi>x</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>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \partial _{2}{\mathcal {L}}(t,x(t),{\dot {x}}(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}(t,x(t),{\dot {x}}(t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a859d35a915e4e25b7e65c56184ee9c10b123df8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:38.317ex; height:3.843ex;" alt="{\textstyle \partial _{2}{\mathcal {L}}(t,x(t),{\dot {x}}(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}(t,x(t),{\dot {x}}(t))}"></span> im gesamten Integrationsintervall gleich null ist (das wird in den Bemerkungen etwas detaillierter erläutert). Damit erhält man für die stationäre Funktion <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> die <i>Euler-Lagrange-Gleichung</i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{2}{\mathcal {L}}(t,x(t),{\dot {x}}(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}(t,x(t),{\dot {x}}(t))=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </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> <mi>t</mi> <mo>,</mo> <mi>x</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>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </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> <mi>t</mi> <mo>,</mo> <mi>x</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>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{2}{\mathcal {L}}(t,x(t),{\dot {x}}(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}(t,x(t),{\dot {x}}(t))=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a317470cf108851fb7ea1eba143014bfe31a9455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.202ex; height:5.509ex;" alt="{\displaystyle \partial _{2}{\mathcal {L}}(t,x(t),{\dot {x}}(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}\partial _{3}{\mathcal {L}}(t,x(t),{\dot {x}}(t))=0}"></span>,</dd></dl> <p>die für alle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in (t_{a},t_{e})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈<!-- ∈ --></mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in (t_{a},t_{e})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b95adc1894db32936edcc219023846972787b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.303ex; height:2.843ex;" alt="{\displaystyle t\in (t_{a},t_{e})}"></span> erfüllt sein muss. </p><p>Die angegebene, zum Verschwinden zu bringende Größe bezeichnet man auch als <i>Eulerableitung</i> der Lagrangefunktion <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 {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\hat {\partial }}{\mathcal {L}}}{{\hat {\partial }}x}}(t):=\left.{\frac {\partial {\mathcal {L}}(t,x,{\dot {x}})}{\partial x}}\right|_{(t,x(t),{\dot {x}}(t))}-{\frac {\mathrm {d} }{\mathrm {d} t}}\,\left(\left.{\frac {\partial {\mathcal {L}}(t,x,{\dot {x}})}{\partial {\dot {x}}}}\right|_{(t,x(t),{\dot {x}}(t))}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mo stretchy="false">^<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mo stretchy="false">^<!-- ^ --></mo> </mover> </mrow> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</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>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msub> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</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>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msub> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\hat {\partial }}{\mathcal {L}}}{{\hat {\partial }}x}}(t):=\left.{\frac {\partial {\mathcal {L}}(t,x,{\dot {x}})}{\partial x}}\right|_{(t,x(t),{\dot {x}}(t))}-{\frac {\mathrm {d} }{\mathrm {d} t}}\,\left(\left.{\frac {\partial {\mathcal {L}}(t,x,{\dot {x}})}{\partial {\dot {x}}}}\right|_{(t,x(t),{\dot {x}}(t))}\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a83ea97ff77ca7c317a35acc418a97e667640f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.874ex; height:7.509ex;" alt="{\displaystyle {\frac {{\hat {\partial }}{\mathcal {L}}}{{\hat {\partial }}x}}(t):=\left.{\frac {\partial {\mathcal {L}}(t,x,{\dot {x}})}{\partial x}}\right|_{(t,x(t),{\dot {x}}(t))}-{\frac {\mathrm {d} }{\mathrm {d} t}}\,\left(\left.{\frac {\partial {\mathcal {L}}(t,x,{\dot {x}})}{\partial {\dot {x}}}}\right|_{(t,x(t),{\dot {x}}(t))}\right)\,.}"></span></dd></dl> <p>Vor allem in Physikbüchern wird die Ableitung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.\partial _{\alpha }\right|_{\alpha =0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.\partial _{\alpha }\right|_{\alpha =0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4b776fc0144b08fe7e33f3a4a085a0a0e4746e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.55ex; height:3.009ex;" alt="{\displaystyle \left.\partial _{\alpha }\right|_{\alpha =0}}"></span> als Variation bezeichnet. Dann ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x=\left.\partial _{\alpha }x_{\alpha }\right|_{\alpha =0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mi>x</mi> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta x=\left.\partial _{\alpha }x_{\alpha }\right|_{\alpha =0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14159df1da7363109e384ee7b3d39cc1d9a18a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.641ex; height:3.009ex;" alt="{\displaystyle \delta x=\left.\partial _{\alpha }x_{\alpha }\right|_{\alpha =0}}"></span> die Variation von <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>. Die Variation der Wirkung </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta I(x,\delta x)=\int {\frac {\delta I}{\delta x(t)}}\delta x(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>δ<!-- δ --></mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ<!-- δ --></mi> <mi>I</mi> </mrow> <mrow> <mi>δ<!-- δ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>δ<!-- δ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta I(x,\delta x)=\int {\frac {\delta I}{\delta x(t)}}\delta x(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef9cd452cf98e305c721f6af084817415b3cbba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.861ex; height:6.176ex;" alt="{\displaystyle \delta I(x,\delta x)=\int {\frac {\delta I}{\delta x(t)}}\delta x(t)\,\mathrm {d} t}"></span></dd></dl> <p>ist wie bei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {d} f=\sum _{i}(\partial _{i}f)\mathrm {d} x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>f</mi> <mo>=</mo> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathrm {d} f=\sum _{i}(\partial _{i}f)\mathrm {d} x^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde6382fd88611f2c801a37927a963f4cd750daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.467ex; height:3.176ex;" alt="{\textstyle \mathrm {d} f=\sum _{i}(\partial _{i}f)\mathrm {d} x^{i}}"></span> eine Linearform in den Variationen der Argumente, ihre Koeffizienten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\delta I}{\delta x(t)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ<!-- δ --></mi> <mi>I</mi> </mrow> <mrow> <mi>δ<!-- δ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\delta I}{\delta x(t)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7510073153eb903680f2307c58ba51e643e39694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.391ex; height:4.343ex;" alt="{\textstyle {\frac {\delta I}{\delta x(t)}}}"></span> heißen <a href="/wiki/Variationsableitung" class="mw-redirect" title="Variationsableitung">Variationsableitung</a> des Funktionals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>. Sie ist im betrachteten Fall die Eulerableitung der Lagrangefunktion </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta I}{\delta x(t)}}={\frac {{\hat {\partial }}{\mathcal {L}}}{{\hat {\partial }}x}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ<!-- δ --></mi> <mi>I</mi> </mrow> <mrow> <mi>δ<!-- δ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mo stretchy="false">^<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mo stretchy="false">^<!-- ^ --></mo> </mover> </mrow> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta I}{\delta x(t)}}={\frac {{\hat {\partial }}{\mathcal {L}}}{{\hat {\partial }}x}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf66ea9de58d7ba6870e711a3e3e933e63bf348e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.539ex; height:6.843ex;" alt="{\displaystyle {\frac {\delta I}{\delta x(t)}}={\frac {{\hat {\partial }}{\mathcal {L}}}{{\hat {\partial }}x}}(t)}"></span>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Bemerkungen">Bemerkungen</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=11" title="Abschnitt bearbeiten: Bemerkungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Bemerkungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Variationsrechnung_hut_polynom6Ordnung.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Variationsrechnung_hut_polynom6Ordnung.png/300px-Variationsrechnung_hut_polynom6Ordnung.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Variationsrechnung_hut_polynom6Ordnung.png/450px-Variationsrechnung_hut_polynom6Ordnung.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Variationsrechnung_hut_polynom6Ordnung.png/600px-Variationsrechnung_hut_polynom6Ordnung.png 2x" data-file-width="640" data-file-height="480" /></a><figcaption>Die Funktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto b(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">↦<!-- ↦ --></mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\mapsto b(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/488632a76f7b7fe2197a343abb118a63a531c878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.1ex; height:2.843ex;" alt="{\displaystyle t\mapsto b(t)}"></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad9cba57d757e960f326f41df30647672e5438c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.155ex; height:2.509ex;" alt="{\displaystyle t_{0}=1}"></span> und <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 \epsilon =0{,}1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ<!-- ϵ --></mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon =0{,}1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43957818545317cf66e9c3cffa22a5fdfa3b523e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.014ex; height:2.509ex;" alt="{\displaystyle \epsilon =0{,}1}"></span></figcaption></figure> <p>Bei der Herleitung der Euler-Lagrange-Gleichung wurde berücksichtigt, dass eine stetige Funktion <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, die für alle mindestens zweimal stetig differenzierbaren Funktionen <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> mit <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 \,b(t_{a})=b(t_{e})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>b</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,b(t_{a})=b(t_{e})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a999545ce4a3af82073c60b649c91b896ccb9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.14ex; height:2.843ex;" alt="{\displaystyle \,b(t_{a})=b(t_{e})=0}"></span> bei Integration über </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 \int _{t_{a}}^{t_{e}}a(t)b(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </msubsup> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{t_{a}}^{t_{e}}a(t)b(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c60d8fa1b1c887215bdfac329d5ffa6793b03ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.508ex; height:6.509ex;" alt="{\displaystyle \int _{t_{a}}^{t_{e}}a(t)b(t)\,\mathrm {d} t}"></span></dd></dl> <p>den Wert null ergibt, identisch gleich null sein muss. </p><p>Das ist leicht einzusehen, wenn man berücksichtigt, dass es zum Beispiel mit </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 b(t):={\begin{cases}0&{\text{für }}t\leq t_{0}-\epsilon {\text{ oder }}t\geq t_{0}+\epsilon \\(t-t_{0}+\epsilon )^{3}(t_{0}-t+\epsilon )^{3}&{\text{für }}t\in (t_{0}-\epsilon ,t_{0}+\epsilon )\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>für </mtext> </mrow> <mi>t</mi> <mo>≤<!-- ≤ --></mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ϵ<!-- ϵ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> oder </mtext> </mrow> <mi>t</mi> <mo>≥<!-- ≥ --></mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ϵ<!-- ϵ --></mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−<!-- − --></mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ϵ<!-- ϵ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>t</mi> <mo>+</mo> <mi>ϵ<!-- ϵ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>für </mtext> </mrow> <mi>t</mi> <mo>∈<!-- ∈ --></mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ϵ<!-- ϵ --></mi> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ϵ<!-- ϵ --></mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b(t):={\begin{cases}0&{\text{für }}t\leq t_{0}-\epsilon {\text{ oder }}t\geq t_{0}+\epsilon \\(t-t_{0}+\epsilon )^{3}(t_{0}-t+\epsilon )^{3}&{\text{für }}t\in (t_{0}-\epsilon ,t_{0}+\epsilon )\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e2cfda7dd5ecc0b939de43a4bdc0724827a56b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.197ex; height:7.509ex;" alt="{\displaystyle b(t):={\begin{cases}0&{\text{für }}t\leq t_{0}-\epsilon {\text{ oder }}t\geq t_{0}+\epsilon \\(t-t_{0}+\epsilon )^{3}(t_{0}-t+\epsilon )^{3}&{\text{für }}t\in (t_{0}-\epsilon ,t_{0}+\epsilon )\end{cases}}}"></span></dd></dl> <p>eine zweimal stetig differenzierbare Funktion gibt, die in einer <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 \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ<!-- ϵ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"></span>-Umgebung eines willkürlich herausgegriffenen Zeitpunktes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}\in (t_{a},t_{e})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈<!-- ∈ --></mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}\in (t_{a},t_{e})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/403e3d50a30e26d480c9aa716f769588100ef685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.357ex; height:2.843ex;" alt="{\displaystyle t_{0}\in (t_{a},t_{e})}"></span> positiv und ansonsten null ist. Gäbe es eine Stelle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}"></span>, an der die Funktion <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> größer oder kleiner null wäre, so wäre sie aufgrund der Stetigkeit auch noch in einer ganzen Umgebung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t_{0}-\epsilon ,t_{0}+\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ϵ<!-- ϵ --></mi> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ϵ<!-- ϵ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t_{0}-\epsilon ,t_{0}+\epsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6018b4653022ad6133240212a57717787ac58d73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.2ex; height:2.843ex;" alt="{\displaystyle (t_{0}-\epsilon ,t_{0}+\epsilon )}"></span> dieser Stelle größer bzw. kleiner null. Mit der eben definierten Funktion <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> ist dann jedoch das Integral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int \limits _{t_{a}}^{t_{b}}a(t)b(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> </munderover> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int \limits _{t_{a}}^{t_{b}}a(t)b(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7738c2fa67c6990dfe17f6833252c8fa53158cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-left: -0.176ex; width:12.214ex; height:6.843ex;" alt="{\textstyle \int \limits _{t_{a}}^{t_{b}}a(t)b(t)\,\mathrm {d} t}"></span> im Widerspruch zur Voraussetzung an <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> ebenfalls größer bzw. kleiner null. Die Annahme, dass <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> an einer Stelle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}"></span> ungleich null wäre, ist also falsch. Die Funktion <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> ist also wirklich identisch gleich null. </p><p>Ist der Funktionenraum <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> ein <a href="/wiki/Affiner_Raum" title="Affiner Raum">affiner Raum</a>, so wird die Familie <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_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>∈<!-- ∈ --></mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>ϵ<!-- ϵ --></mi> <mo>,</mo> <mi>ϵ<!-- ϵ --></mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500983860ea85f72043d1716a91dc513eb4b92e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.154ex; height:3.176ex;" alt="{\displaystyle (x_{\alpha })_{\alpha \in (-\epsilon ,\epsilon )}}"></span> in der Literatur oftmals als Summe <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_{\alpha }(t):=x(t)+\alpha h(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>α<!-- α --></mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\alpha }(t):=x(t)+\alpha h(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8356587640cddd74cb12c3fd66ab2cf2de9a7095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.303ex; height:2.843ex;" alt="{\displaystyle x_{\alpha }(t):=x(t)+\alpha h(t)}"></span> mit einer frei wählbaren Zeitfunktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> festgelegt, die der Bedingung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t_{a})=h(t_{e})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(t_{a})=h(t_{e})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a2d99064b25534e5ea1c43f1a6ebaa18468077e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.435ex; height:2.843ex;" alt="{\displaystyle h(t_{a})=h(t_{e})=0}"></span> genügen muss. Die Ableitung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.\partial _{\alpha }I(x_{\alpha })\right|_{\alpha =0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.\partial _{\alpha }I(x_{\alpha })\right|_{\alpha =0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c38bab5ffb5bfd7ad504feb7c03728e9b163d65a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.145ex; height:3.009ex;" alt="{\displaystyle \left.\partial _{\alpha }I(x_{\alpha })\right|_{\alpha =0}}"></span> ist dann gerade die <a href="/wiki/Gateaux-Ableitung" class="mw-redirect" title="Gateaux-Ableitung">Gateaux-Ableitung</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 \left.\partial _{\alpha }I(x+\alpha h)\right|_{\alpha =0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msub> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>α<!-- α --></mi> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.\partial _{\alpha }I(x+\alpha h)\right|_{\alpha =0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b001c12776b1a88e4e301505eb5157ee8134defa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.528ex; height:3.009ex;" alt="{\displaystyle \left.\partial _{\alpha }I(x+\alpha h)\right|_{\alpha =0}}"></span> des Funktionals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> an der Stelle <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> in Richtung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>. Die hier vorgestellte Version erscheint dem Autor etwas günstiger, wenn die Funktionenmenge <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> kein affiner Raum mehr ist (wenn sie beispielsweise durch eine nichtlineare Nebenbedingung eingeschränkt ist; siehe etwa <a href="/wiki/Prinzip_des_kleinsten_Zwanges" title="Prinzip des kleinsten Zwanges">gaußsches Prinzip des kleinsten Zwanges</a>). Sie ist ausführlicher bei Smirnow<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> dargestellt und lehnt sich an die Definition von <a href="/wiki/Tangentialvektor" class="mw-redirect" title="Tangentialvektor">Tangentialvektoren</a> an Mannigfaltigkeiten an.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>Im Falle eines weiteren, einschränkenden Funktionals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle J(x)=\int j(t,x,{\dot {x}},{\ddot {x}},\dots ,x^{(n)})d^{d}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫<!-- ∫ --></mo> <mi>j</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙<!-- ˙ --></mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨<!-- ¨ --></mo> </mover> </mrow> </mrow> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle J(x)=\int j(t,x,{\dot {x}},{\ddot {x}},\dots ,x^{(n)})d^{d}t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/663835cfa1312b8891747ebab45b763162678519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.367ex; height:3.343ex;" alt="{\textstyle J(x)=\int j(t,x,{\dot {x}},{\ddot {x}},\dots ,x^{(n)})d^{d}t}"></span>, der den Funktionenraum <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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> dadurch einschränkt, dass <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(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a1ea17a6ba74ffe226bca76db3da3b66616174c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.871ex; height:2.843ex;" alt="{\displaystyle J(x)=0}"></span> gelten soll, kann man analog zum reellen Fall das Verfahren der <a href="/wiki/Lagrange-Multiplikator" title="Lagrange-Multiplikator">Lagrange-Multiplikatoren</a> anwenden: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta I}{\delta x_{i}}}=\lambda {\frac {\delta J}{\delta x_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ<!-- δ --></mi> <mi>I</mi> </mrow> <mrow> <mi>δ<!-- δ --></mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ<!-- δ --></mi> <mi>J</mi> </mrow> <mrow> <mi>δ<!-- δ --></mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta I}{\delta x_{i}}}=\lambda {\frac {\delta J}{\delta x_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb56b2076f90cb95fcf8dcf996c2536284286df2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.482ex; height:5.843ex;" alt="{\displaystyle {\frac {\delta I}{\delta x_{i}}}=\lambda {\frac {\delta J}{\delta x_{i}}}}"></span></dd></dl> <p>für beliebiges <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dotsc ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dotsc ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f2132430a61b900cf2c4380774394ca9f09c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.636ex; height:2.509ex;" alt="{\displaystyle i=1,\dotsc ,n}"></span> und ein festes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ<!-- λ --></mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87bc1622689bc998795834cd65eecdb4955a785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.874ex; height:2.176ex;" alt="{\displaystyle \lambda \in \mathbb {R} }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Verallgemeinerung_für_höhere_Ableitung_und_Dimensionen"><span id="Verallgemeinerung_f.C3.BCr_h.C3.B6here_Ableitung_und_Dimensionen"></span>Verallgemeinerung für höhere Ableitung und Dimensionen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=12" title="Abschnitt bearbeiten: Verallgemeinerung für höhere Ableitung und Dimensionen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Verallgemeinerung für höhere Ableitung und Dimensionen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die obige Herleitung mittels partieller Integration lässt sich auf Variationsprobleme der Art </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 I(\varphi )=\int {\mathcal {L}}(\varphi (x),\partial _{1}\varphi (x),\dots ,\partial _{d}\varphi (x),\dots )\mathrm {d} ^{d}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>φ<!-- φ --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫<!-- ∫ --></mo> <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>φ<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>φ<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mi>φ<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…<!-- … --></mo> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\varphi )=\int {\mathcal {L}}(\varphi (x),\partial _{1}\varphi (x),\dots ,\partial _{d}\varphi (x),\dots )\mathrm {d} ^{d}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e61cb56d494b73dc7a6acb95a089dae4e48b03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.87ex; height:5.676ex;" alt="{\displaystyle I(\varphi )=\int {\mathcal {L}}(\varphi (x),\partial _{1}\varphi (x),\dots ,\partial _{d}\varphi (x),\dots )\mathrm {d} ^{d}x}"></span></dd></dl> <p>übertragen, wobei in den Abhängigkeiten Ableitungen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{\alpha }\varphi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mi>φ<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{\alpha }\varphi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66db02210011de46292c4970a1fdf62f8445f9c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.868ex; height:2.843ex;" alt="{\displaystyle D^{\alpha }\varphi (x)}"></span> (siehe <a href="/wiki/Multiindex" title="Multiindex">Multiindex-Notation</a>) auch höherer Ordnung auftauchen, etwa bis zur Ordnung <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 \vert \alpha \vert \leq N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>α<!-- α --></mi> <mo fence="false" stretchy="false">|</mo> <mo>≤<!-- ≤ --></mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert \alpha \vert \leq N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec5d67eaf5d94b97aa44437e688a13006e597ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.943ex; height:2.843ex;" alt="{\displaystyle \vert \alpha \vert \leq N}"></span>. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/346ffaf97bb2e947e9dfcb633d99001b4b191254" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.208ex; height:2.343ex;" alt="{\displaystyle D^{\alpha }}"></span> ist gerade der <a href="/wiki/Differentialoperator" title="Differentialoperator">Differentialoperator</a>. In diesem Fall lautet die Euler-Lagrange-Gleichung: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\vert \alpha \vert \leq N}(-1)^{\vert \alpha \vert }D^{\alpha }{\frac {\delta {\mathcal {L}}}{\delta (D^{\alpha }\varphi (x))}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">|</mo> <mi>α<!-- α --></mi> <mo fence="false" stretchy="false">|</mo> <mo>≤<!-- ≤ --></mo> <mi>N</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">|</mo> <mi>α<!-- α --></mi> <mo fence="false" stretchy="false">|</mo> </mrow> </msup> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi>δ<!-- δ --></mi> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mi>φ<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\vert \alpha \vert \leq N}(-1)^{\vert \alpha \vert }D^{\alpha }{\frac {\delta {\mathcal {L}}}{\delta (D^{\alpha }\varphi (x))}}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e26b0142876667428eeaa9f95a44eb8f7742861" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:31.361ex; height:7.009ex;" alt="{\displaystyle \sum _{\vert \alpha \vert \leq N}(-1)^{\vert \alpha \vert }D^{\alpha }{\frac {\delta {\mathcal {L}}}{\delta (D^{\alpha }\varphi (x))}}=0,}"></span></dd></dl> <p>wobei die Euler-Ableitung als </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta {\mathcal {L}}}{\delta (D^{\alpha }\varphi (x))}}:=\left.{\frac {\partial {\mathcal {L}}}{\partial (D^{\alpha }\varphi )}}\right\vert _{\varphi =\varphi (x),\partial _{1}\varphi =\partial _{1}\varphi (x),\dots }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi>δ<!-- δ --></mi> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mi>φ<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>:=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mi>φ<!-- φ --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> <mo>=</mo> <mi>φ<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>φ<!-- φ --></mi> <mo>=</mo> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>φ<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…<!-- … --></mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta {\mathcal {L}}}{\delta (D^{\alpha }\varphi (x))}}:=\left.{\frac {\partial {\mathcal {L}}}{\partial (D^{\alpha }\varphi )}}\right\vert _{\varphi =\varphi (x),\partial _{1}\varphi =\partial _{1}\varphi (x),\dots }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/426bf862b8eb7001fc34bc17e5550ad12dcc7120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.423ex; height:7.009ex;" alt="{\displaystyle {\frac {\delta {\mathcal {L}}}{\delta (D^{\alpha }\varphi (x))}}:=\left.{\frac {\partial {\mathcal {L}}}{\partial (D^{\alpha }\varphi )}}\right\vert _{\varphi =\varphi (x),\partial _{1}\varphi =\partial _{1}\varphi (x),\dots }}"></span></dd></dl> <p>zu verstehen ist. Hierbei sind in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{\alpha }\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{\alpha }\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965a0bf1ffb09550ead4c96bf0fa817cf30bdb90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.729ex; height:2.843ex;" alt="{\displaystyle D^{\alpha }\varphi }"></span> in selbsterklärender Weise symbolisch die entsprechende Abhängigkeit von <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 {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span> repräsentiert, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{\alpha }\varphi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mi>φ<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{\alpha }\varphi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66db02210011de46292c4970a1fdf62f8445f9c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.868ex; height:2.843ex;" alt="{\displaystyle D^{\alpha }\varphi (x)}"></span> steht für den konkreten Wert der Ableitung von <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 \varphi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4046f1f2de7df04bde418ba2bc4d3898ac2385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.659ex; height:2.843ex;" alt="{\displaystyle \varphi (x)}"></span>. Insbesondere wird auch über <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cc00f65bbc630448311dd2dc82e7ce5e90985a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =0}"></span> summiert. </p> <div class="mw-heading mw-heading3"><h3 id="Weiterführende_Verallgemeinerungen"><span id="Weiterf.C3.BChrende_Verallgemeinerungen"></span>Weiterführende Verallgemeinerungen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=13" title="Abschnitt bearbeiten: Weiterführende Verallgemeinerungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Weiterführende Verallgemeinerungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Verallgemeinerungen für mehrere Funktionen, Dimensionen und auf <a href="/wiki/Mannigfaltigkeit" title="Mannigfaltigkeit">Mannigfaltigkeiten</a> können gemacht werden. In diesem Zusammenhang ist es günstig, einen sog. <a href="/w/index.php?title=Euler-Operator&action=edit&redlink=1" class="new" title="Euler-Operator (Seite nicht vorhanden)">Euler-Operator</a> einzuführen und davon gebrauch zu machen, wobei verschiedene Ansätze für den Operator existieren.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=14" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Siehe auch"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Banachraum" title="Banachraum">Banachraum</a></li> <li><a href="/wiki/Ginsburg-Landau-Theorie" title="Ginsburg-Landau-Theorie">Ginzburg-Landau-Theorie</a> (vgl. auch <a href="/wiki/Landau-Theorie" title="Landau-Theorie">Landau-Theorie</a>)</li> <li><a href="/wiki/Hamilton-Jacobi-Formalismus" title="Hamilton-Jacobi-Formalismus">Hamilton-Jacobi-Formalismus</a></li> <li><a href="/wiki/Schwingers_Quantenwirkungsprinzip" title="Schwingers Quantenwirkungsprinzip">Schwingers Quantenwirkungsprinzip</a></li> <li><a href="/wiki/Variation_der_Elemente" title="Variation der Elemente">Variation der Elemente</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=15" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Journale_&_andere_Beiträge"><span id="Journale_.26_andere_Beitr.C3.A4ge"></span>Journale & andere Beiträge</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=16" title="Abschnitt bearbeiten: Journale & andere Beiträge" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Journale & andere Beiträge"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.springer.com/journal/526">Calculus of Variations and Partial Differential Equations</a> (<span class="plainlinks-print"><a href="/wiki/Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%221432-0835%22&key=cql">1432-0835</a></span></span>)</li> <li><a rel="nofollow" class="external text" href="https://www.degruyter.com/journal/key/acv/html">Advances in Calculus of Variations</a> (<span class="plainlinks-print"><a href="/wiki/Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%221864-8266%22&key=cql">1864-8266</a></span></span>)</li> <li>Alessio Figalli, Robert V. Kohn, Tatiana Toro, Neshan Wickramasekera: <cite style="font-style:italic">Calculus of Variations</cite>. In: <cite style="font-style:italic">Oberwolfach Reports</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>17</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>2</span>, 1. Juli 2021, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>1139–1196</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.4171/owr%2F2020%2F22">10.4171/owr/2020/22</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=Calculus+of+Variations&rft.au=Alessio+Figalli%2C+Robert+V.+Kohn%2C+Tatiana+Toro%2C+...&rft.date=2021-07-01&rft.doi=10.4171%2Fowr%2F2020%2F22&rft.genre=journal&rft.issue=2&rft.jtitle=Oberwolfach+Reports&rft.pages=1139-1196&rft.volume=17" style="display:none"> </span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Schools_&_Workshops"><span id="Schools_.26_Workshops"></span>Schools & Workshops</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=17" title="Abschnitt bearbeiten: Schools & Workshops" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=17" title="Quellcode des Abschnitts bearbeiten: Schools & Workshops"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://appliedmath.univie.ac.at/public/second_austrian_calculus_of_variations_day/">2nd Austrian Calculus of Variations Day</a> (2022) – <a href="/wiki/Universit%C3%A4t_Wien" title="Universität Wien">Universität Wien</a></li> <li><a rel="nofollow" class="external text" href="https://sites.google.com/unifi.it/advcalcvar2022/home-page">Advances in Calculus of Variations</a> (2022) – Verschiedene Sponsoren</li> <li><a rel="nofollow" class="external text" href="https://sites.google.com/view/tcvpde-2022/home">Trends in Calculus of Variations and PDEs</a> (2022) – <a href="/wiki/University_of_Sussex" title="University of Sussex">University of Sussex</a>, <a href="/wiki/Universit%C3%A4t_Gent" title="Universität Gent">Universität Gent</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Skripte">Skripte</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=18" title="Abschnitt bearbeiten: Skripte" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=18" title="Quellcode des Abschnitts bearbeiten: Skripte"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Andreas Klaiber: <cite style="font-style:italic">Variationsrechnung</cite>. 2016 (<a rel="nofollow" class="external text" href="https://www.math.uni-konstanz.de/~klaiber/Skript_Variationsrechnung.pdf">uni-konstanz.de</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Andreas+Klaiber&rft.btitle=Variationsrechnung&rft.date=2016&rft.genre=book" style="display:none"> </span></li> <li>Erich Miersemann: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Calculus of Variations</cite>. 2021 (englisch, <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=84992018520e1ee2e6ec40440b5853e6a5701495">psu.edu</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Erich+Miersemann&rft.btitle=Calculus+of+Variations&rft.date=2021&rft.genre=book" style="display:none"> </span></li> <li><a href="/wiki/Peter_J._Olver" class="mw-redirect" title="Peter J. Olver">Peter J. Olver</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">The Calculus of Variations</cite>. 2022 (englisch, <a rel="nofollow" class="external text" href="https://www-users.cse.umn.edu/~olver/ln_/cvc.pdf">umn.edu</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Peter+J.+Olver&rft.btitle=The+Calculus+of+Variations&rft.date=2022&rft.genre=book" style="display:none"> </span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=19" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=19" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Moderne_Lehrbücher"><span id="Moderne_Lehrb.C3.BCcher"></span>Moderne Lehrbücher</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=20" title="Abschnitt bearbeiten: Moderne Lehrbücher" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=20" title="Quellcode des Abschnitts bearbeiten: Moderne Lehrbücher"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Hansjörg Kielhöfer: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Calculus of Variations</cite> (= <cite class="lang" lang="en" dir="auto" style="font-style:italic">Texts in Applied Mathematics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>67</span>). Springer International Publishing, Cham 2018, <a href="/wiki/Spezial:ISBN-Suche/9783319711225" class="internal mw-magiclink-isbn">ISBN 978-3-319-71122-5</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-319-71123-2">10.1007/978-3-319-71123-2</a></span> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Hansj%C3%B6rg+Kielh%C3%B6fer&rft.btitle=Calculus+of+Variations&rft.date=2018&rft.doi=10.1007%2F978-3-319-71123-2&rft.genre=book&rft.isbn=9783319711225&rft.place=Cham&rft.pub=Springer+International+Publishing&rft.series=Texts+in+Applied+Mathematics" style="display:none"> </span></li> <li>Francis Clarke: <cite style="font-style:italic">Functional Analysis, Calculus of Variations and Optimal Control</cite> (= <cite style="font-style:italic"><a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a></cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>264</span>). Springer, London 2013, <a href="/wiki/Spezial:ISBN-Suche/9781447148197" class="internal mw-magiclink-isbn">ISBN 978-1-4471-4819-7</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-1-4471-4820-3">10.1007/978-1-4471-4820-3</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Francis+Clarke&rft.btitle=Functional+Analysis%2C+Calculus+of+Variations+and+Optimal+Control&rft.date=2013&rft.doi=10.1007%2F978-1-4471-4820-3&rft.genre=book&rft.isbn=9781447148197&rft.place=London&rft.pub=Springer&rft.series=Graduate+Texts+in+Mathematics" style="display:none"> </span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Monografien">Monografien</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=21" title="Abschnitt bearbeiten: Monografien" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=21" title="Quellcode des Abschnitts bearbeiten: Monografien"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Philippe Blanchard, Erwin Brüning: <cite style="font-style:italic">Direkte Methoden der Variationsrechnung</cite>. Springer, Vienna 1982, <a href="/wiki/Spezial:ISBN-Suche/9783709122617" class="internal mw-magiclink-isbn">ISBN 978-3-7091-2261-7</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-7091-2260-0">10.1007/978-3-7091-2260-0</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Philippe+Blanchard%2C+Erwin+Br%C3%BCning&rft.btitle=Direkte+Methoden+der+Variationsrechnung&rft.date=1982&rft.doi=10.1007%2F978-3-7091-2260-0&rft.genre=book&rft.isbn=9783709122617&rft.place=Vienna&rft.pub=Springer" style="display:none"> </span></li> <li><a href="/wiki/Mariano_Giaquinta" title="Mariano Giaquinta">Mariano Giaquinta</a>, <a href="/wiki/Stefan_Hildebrandt" title="Stefan Hildebrandt">Stefan Hildebrandt</a>: <cite style="font-style:italic">Calculus of Variations I</cite> (= A. Chenciner u. a. [Hrsg.]: <cite style="font-style:italic"><a href="/wiki/Grundlehren_der_mathematischen_Wissenschaften" title="Grundlehren der mathematischen Wissenschaften">Grundlehren der mathematischen Wissenschaften</a></cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>310</span>). Springer, Berlin/Heidelberg 2004, <a href="/wiki/Spezial:ISBN-Suche/9783642080746" class="internal mw-magiclink-isbn">ISBN 978-3-642-08074-6</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-662-03278-7">10.1007/978-3-662-03278-7</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Mariano+Giaquinta%2C+Stefan+Hildebrandt&rft.btitle=Calculus+of+Variations+I&rft.date=2004&rft.doi=10.1007%2F978-3-662-03278-7&rft.genre=book&rft.isbn=9783642080746&rft.place=Berlin%2FHeidelberg&rft.pub=Springer&rft.series=Grundlehren+der+mathematischen+Wissenschaften" style="display:none"> </span></li> <li><a href="/wiki/Mariano_Giaquinta" title="Mariano Giaquinta">Mariano Giaquinta</a>, <a href="/wiki/Stefan_Hildebrandt" title="Stefan Hildebrandt">Stefan Hildebrandt</a>: <cite style="font-style:italic">Calculus of Variations II</cite> (= A. Chenciner u. a. [Hrsg.]: <cite style="font-style:italic"><a href="/wiki/Grundlehren_der_mathematischen_Wissenschaften" title="Grundlehren der mathematischen Wissenschaften">Grundlehren der mathematischen Wissenschaften</a></cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>311</span>). Springer, Berlin/Heidelberg 2004, <a href="/wiki/Spezial:ISBN-Suche/9783642081927" class="internal mw-magiclink-isbn">ISBN 978-3-642-08192-7</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-662-06201-2">10.1007/978-3-662-06201-2</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Mariano+Giaquinta%2C+Stefan+Hildebrandt&rft.btitle=Calculus+of+Variations+II&rft.date=2004&rft.doi=10.1007%2F978-3-662-06201-2&rft.genre=book&rft.isbn=9783642081927&rft.place=Berlin%2FHeidelberg&rft.pub=Springer&rft.series=Grundlehren+der+mathematischen+Wissenschaften" style="display:none"> </span></li> <li><a href="/wiki/J%C3%BCrgen_Jost" title="Jürgen Jost">Jürgen Jost</a>, Xianqing Li-Jost: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Calculus of variations</cite> (= <cite class="lang" lang="en" dir="auto" style="font-style:italic">Cambridge studies in advanced mathematics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>64</span>). 1. publ Auflage. Cambridge Univ. Press, Cambridge 1998, <a href="/wiki/Spezial:ISBN-Suche/9780521057127" class="internal mw-magiclink-isbn">ISBN 978-0-521-05712-7</a> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=J%C3%BCrgen+Jost%2C+Xianqing+Li-Jost&rft.btitle=Calculus+of+variations&rft.date=1998&rft.edition=1.+publ&rft.genre=book&rft.isbn=9780521057127&rft.place=Cambridge&rft.pub=Cambridge+Univ.+Press&rft.series=Cambridge+studies+in+advanced+mathematics" style="display:none"> </span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Klassische_und_historische_Werke">Klassische und historische Werke</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=22" title="Abschnitt bearbeiten: Klassische und historische Werke" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=22" title="Quellcode des Abschnitts bearbeiten: Klassische und historische Werke"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Oskar_Bolza" title="Oskar Bolza">Oskar Bolza</a>: <i>Vorlesungen über Variationsrechnung.</i> B. G. Teubner, Leipzig u. a. 1909, (<a href="//archive.org/details/vorlesungenuberv028879mbp" class="extiw" title="iarchive:vorlesungenuberv028879mbp">Digitalisat</a>). Dover 2018 (englisch).</li> <li><a href="/wiki/Paul_Funk_(Mathematiker)" title="Paul Funk (Mathematiker)">Paul Funk</a>: <cite style="font-style:italic">Variationsrechnung und ihre Anwendung in Physik und Technik</cite>. Springer, Berlin/Heidelberg 1970, <a href="/wiki/Spezial:ISBN-Suche/9783642885983" class="internal mw-magiclink-isbn">ISBN 978-3-642-88598-3</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-642-88597-6">10.1007/978-3-642-88597-6</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Paul+Funk&rft.btitle=Variationsrechnung+und+ihre+Anwendung+in+Physik+und+Technik&rft.date=1970&rft.doi=10.1007%2F978-3-642-88597-6&rft.genre=book&rft.isbn=9783642885983&rft.place=Berlin%2FHeidelberg&rft.pub=Springer" style="display:none"> </span></li> <li><a href="/wiki/Israel_Moissejewitsch_Gelfand" title="Israel Moissejewitsch Gelfand">I. M. Gelfand</a>, <a href="/wiki/Sergei_Wassiljewitsch_Fomin" title="Sergei Wassiljewitsch Fomin">S. W. Fomin</a>: <cite style="font-style:italic">Calculus of variations</cite>. Dover Publications, Mineola, NY 2000, <a href="/wiki/Spezial:ISBN-Suche/9780486414485" class="internal mw-magiclink-isbn">ISBN 978-0-486-41448-5</a> (Originaltitel: <cite style="font-style:italic">Calculus of variations</cite>. 1963. Übersetzt von Richard A. Silverman).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=I.+M.+Gelfand%2C+S.+W.+Fomin&rft.btitle=Calculus+of+variations&rft.date=2000&rft.genre=book&rft.isbn=9780486414485&rft.place=Mineola%2C+NY&rft.pub=Dover+Publications" style="display:none"> </span></li> <li><a href="/wiki/Adolf_Kneser" title="Adolf Kneser">Adolf Kneser</a>: <i>Variationsrechnung.</i> In: <i><a href="/wiki/Encyklop%C3%A4die_der_mathematischen_Wissenschaften_mit_Einschluss_ihrer_Anwendungen" class="mw-redirect" title="Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen">Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen</a>.</i> Band 2: <i>Analysis.</i> Teil 1. B. G. Teubner, Leipzig 1898, <a rel="nofollow" class="external text" href="https://gdz.sub.uni-goettingen.de/id/PPN36050616X?tify=%7B%22pages%22%3A%5B593%5D%2C%22pan%22%3A%7B%22x%22%3A0.416%2C%22y%22%3A0.828%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.301%7D">S. 571–625</a>.</li> <li><a href="/wiki/Solomon_Grigorjewitsch_Michlin" title="Solomon Grigorjewitsch Michlin">S. G. Michlin</a>: <cite style="font-style:italic">Variationsmethoden der mathematischen Physik</cite>. Akademie-Verlag, Berlin 1962.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=S.+G.+Michlin&rft.btitle=Variationsmethoden+der+mathematischen+Physik&rft.date=1962&rft.genre=book&rft.place=Berlin&rft.pub=Akademie-Verlag" style="display:none"> </span></li> <li><a href="/wiki/Paul_St%C3%A4ckel" title="Paul Stäckel">Paul Stäckel</a> (Hrsg.): <i>Abhandlungen über Variations-Rechnung.</i> 2 Theile. Wilhelm Engelmann, Leipzig 1894; <ul><li>Theil 1: <i>Abhandlungen von Joh. Bernoulli (1696), Jac. Bernoulli (1697) und Leonhard Euler (1744)</i> (= <i>Ostwald’s Klassiker der exakten Wissenschaften.</i> 46, <span class="plainlinks-print"><a href="/wiki/Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220232-3419%22&key=cql">0232-3419</a></span></span>). 1894, (<a rel="nofollow" class="external text" href="https://archive.org/stream/abhandlungenber01jacogoog#page/n3/mode/2up">Digitalisat</a>);</li> <li>Theil 2: <i>Abhandlungen von Lagrange (1762, 1770), Legendre (1786), und Jacobi (1837)</i> (= <i>Ostwald’s Klassiker der exakten Wissenschaften.</i> 47). 1894, (<a rel="nofollow" class="external text" href="https://archive.org/stream/abhandlungenber02jacogoog#page/n3/mode/2up">Digitalisat</a>).</li></ul></li> <li><a href="/wiki/Friedrich_Stegmann" title="Friedrich Stegmann">Friedrich Stegmann</a>; <i>Lehrbuch der Variationsrechnung und ihrer Anwendung bei Untersuchungen über das Maximum und Minimum.</i> Kassel, Luckhardt, 1854.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Variationsrechnung&veaction=edit&section=23" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Variationsrechnung&action=edit&section=23" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Jeremy Gray: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Change and Variations: A History of Differential Equations to 1900</cite> (= <cite class="lang" lang="en" dir="auto" style="font-style:italic">Springer Undergraduate Mathematics Series</cite>). Springer International Publishing, Cham 2021, <a href="/wiki/Spezial:ISBN-Suche/9783030705749" class="internal mw-magiclink-isbn">ISBN 978-3-03070574-9</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-030-70575-6">10.1007/978-3-030-70575-6</a></span> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Jeremy+Gray&rft.btitle=Change+and+Variations%3A+A+History+of+Differential+Equations+to+1900&rft.date=2021&rft.doi=10.1007%2F978-3-030-70575-6&rft.genre=book&rft.isbn=9783030705749&rft.place=Cham&rft.pub=Springer+International+Publishing&rft.series=Springer+Undergraduate+Mathematics+Series" style="display:none"> </span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite style="font-style:italic">Mathematik für Physiker 2</cite> (= <cite style="font-style:italic">Springer-Lehrbuch</cite>). Springer, Berlin/Heidelberg 2007, <a href="/wiki/Spezial:ISBN-Suche/9783540722519" class="internal mw-magiclink-isbn">ISBN 978-3-540-72251-9</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-540-72252-6">10.1007/978-3-540-72252-6</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.btitle=Mathematik+f%C3%BCr+Physiker+2&rft.date=2007&rft.doi=10.1007%2F978-3-540-72252-6&rft.genre=book&rft.isbn=9783540722519&rft.place=Berlin%2FHeidelberg&rft.pub=Springer&rft.series=Springer-Lehrbuch" style="display:none"> </span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Hubert Goldschmidt, Shlomo Sternberg: <cite style="font-style:italic">The Hamilton-Cartan formalism in the calculus of variations</cite>. In: <cite style="font-style:italic">Annales de l’institut Fourier</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>23</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>1</span>, 1973, <a href="/wiki/Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220373-0956%22&key=cql">0373-0956</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>203–267</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.5802/aif.451">10.5802/aif.451</a></span> (<a rel="nofollow" class="external text" href="https://aif.centre-mersenne.org/item/AIF_1973__23_1_203_0/">centre-mersenne.org</a> [abgerufen am 21. Oktober 2022]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=The+Hamilton-Cartan+formalism+in+the+calculus+of+variations&rft.au=Hubert+Goldschmidt%2C+Shlomo+Sternberg&rft.date=1973&rft.doi=10.5802%2Faif.451&rft.genre=journal&rft.issn=0373-0956&rft.issue=1&rft.jtitle=Annales+de+l%E2%80%99institut+Fourier&rft.pages=203-267&rft.volume=23" style="display:none"> </span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Vladimir I. Pupyshev, H. E. Montgomery: <cite style="font-style:italic">Some problems in applications of the linear variational method</cite>. In: <cite style="font-style:italic">European Journal of Physics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>36</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>5</span>, 1. September 2015, <a href="/wiki/Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220143-0807%22&key=cql">0143-0807</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>055043</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1088/0143-0807%2F36%2F5%2F055043">10.1088/0143-0807/36/5/055043</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=Some+problems+in+applications+of+the+linear+variational+method&rft.au=Vladimir+I.+Pupyshev%2C+H.+E.+Montgomery&rft.date=2015-09-01&rft.doi=10.1088%2F0143-0807%2F36%2F5%2F055043&rft.genre=journal&rft.issn=0143-0807&rft.issue=5&rft.jtitle=European+Journal+of+Physics&rft.pages=055043&rft.volume=36" style="display:none"> </span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">E. Noether, M. A. Tavel: <cite style="font-style:italic">Invariant Variation Problems</cite>. In: <cite style="font-style:italic">Transport Theory and Statistical Physics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>1</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>3</span>, Januar 1971, <a href="/wiki/Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220041-1450%22&key=cql">0041-1450</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>186–207</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1080/00411457108231446">10.1080/00411457108231446</a></span>, <a href="/wiki/ArXiv" title="ArXiv">arxiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/0503066">physics/0503066</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=Invariant+Variation+Problems&rft.au=E.+Noether%2C+M.+A.+Tavel&rft.date=1971-01&rft.doi=10.1080%2F00411457108231446&rft.genre=journal&rft.issn=0041-1450&rft.issue=3&rft.jtitle=Transport+Theory+and+Statistical+Physics&rft.pages=186-207&rft.volume=1" style="display:none"> </span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Philippe Blanchard, Erwin Brüning: <cite style="font-style:italic">Klassische Variationsprobleme</cite>. In: <cite style="font-style:italic">Direkte Methoden der Variationsrechnung</cite>. Springer Vienna, Wien 1982, <a href="/wiki/Spezial:ISBN-Suche/9783709122617" class="internal mw-magiclink-isbn">ISBN 978-3-7091-2261-7</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>74–124</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-7091-2260-0_6">10.1007/978-3-7091-2260-0_6</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=Klassische+Variationsprobleme&rft.au=Philippe+Blanchard%2C+Erwin+Br%C3%BCning&rft.btitle=Direkte+Methoden+der+Variationsrechnung&rft.date=1982&rft.doi=10.1007%2F978-3-7091-2260-0_6&rft.genre=book&rft.isbn=9783709122617&rft.pages=74-124&rft.place=Wien&rft.pub=Springer+Vienna" style="display:none"> </span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://abelprize.no/abel-prize-laureates/2019"><i>The Abel Prize. 2019: Karen Keskulla Uhlenbeck.</i></a><span class="Abrufdatum"> Abgerufen am 18. Oktober 2022</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AVariationsrechnung&rft.title=The+Abel+Prize.+2019%3A+Karen+Keskulla+Uhlenbeck&rft.description=The+Abel+Prize.+2019%3A+Karen+Keskulla+Uhlenbeck&rft.identifier=https%3A%2F%2Fabelprize.no%2Fabel-prize-laureates%2F2019"> </span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">Simon Donaldson: <cite style="font-style:italic">Karen Uhlenbeck and the Calculus of Variations</cite>. In: <cite style="font-style:italic">Notices of the American Mathematical Society</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>66</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>03</span>, 1. März 2019, <a href="/wiki/Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220002-9920%22&key=cql">0002-9920</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>1</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1090/noti1806">10.1090/noti1806</a></span> (<a rel="nofollow" class="external text" href="https://www.ams.org/journals/notices/201903/rnoti-p303.pdf">ams.org</a> [PDF; abgerufen am 18. Oktober 2022]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=Karen+Uhlenbeck+and+the+Calculus+of+Variations&rft.au=Simon+Donaldson&rft.date=2019-03-01&rft.doi=10.1090%2Fnoti1806&rft.genre=journal&rft.issn=0002-9920&rft.issue=03&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.pages=1&rft.volume=66" style="display:none"> </span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><a href="/wiki/Richard_Courant" title="Richard Courant">Richard Courant</a>, <a href="/wiki/Herbert_Robbins" title="Herbert Robbins">Herbert Robbins</a>: <cite style="font-style:italic">Siebentes Kapitel. Maxima und Minima</cite>. In: <cite style="font-style:italic">Was ist Mathematik?</cite> Springer, Berlin/Heidelberg 2001, <a href="/wiki/Spezial:ISBN-Suche/9783642137006" class="internal mw-magiclink-isbn">ISBN 978-3-642-13700-6</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>251–301</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-642-13701-3_7">10.1007/978-3-642-13701-3_7</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=Siebentes+Kapitel.+Maxima+und+Minima&rft.au=Richard+Courant%2C+Herbert+Robbins&rft.btitle=Was+ist+Mathematik%3F&rft.date=2001&rft.doi=10.1007%2F978-3-642-13701-3_7&rft.genre=book&rft.isbn=9783642137006&rft.pages=251-301&rft.place=Berlin%2FHeidelberg&rft.pub=Springer" style="display:none"> </span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Weierstrass_conditions_(for_a_variational_extremum)"><i>Weierstrass conditions (for a variational extremum) – Encyclopedia of Mathematics.</i></a><span class="Abrufdatum"> Abgerufen am 19. Oktober 2022</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AVariationsrechnung&rft.title=Weierstrass+conditions+%28for+a+variational+extremum%29+%E2%80%93+Encyclopedia+of+Mathematics&rft.description=Weierstrass+conditions+%28for+a+variational+extremum%29+%E2%80%93+Encyclopedia+of+Mathematics&rft.identifier=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FWeierstrass_conditions_%28for_a_variational_extremum%29"> </span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Weierstrass-Erdmann_corner_conditions"><i>Weierstrass-Erdmann corner conditions – Encyclopedia of Mathematics.</i></a><span class="Abrufdatum"> Abgerufen am 19. Oktober 2022</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AVariationsrechnung&rft.title=Weierstrass-Erdmann+corner+conditions+%E2%80%93+Encyclopedia+of+Mathematics&rft.description=Weierstrass-Erdmann+corner+conditions+%E2%80%93+Encyclopedia+of+Mathematics&rft.identifier=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FWeierstrass-Erdmann_corner_conditions"> </span></span> </li> <li id="cite_note-:0-12"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:0_12-0">a</a></sup> <sup><a href="#cite_ref-:0_12-1">b</a></sup> <sup><a href="#cite_ref-:0_12-2">c</a></sup> <sup><a href="#cite_ref-:0_12-3">d</a></sup> <sup><a href="#cite_ref-:0_12-4">e</a></sup></span> <span class="reference-text">Hansjörg Kielhöfer: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Calculus of Variations</cite> (= <cite class="lang" lang="en" dir="auto" style="font-style:italic">Texts in Applied Mathematics</cite>. Texts in Applied Mathematics). <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>67</span>. Springer International Publishing, Cham 2018, <a href="/wiki/Spezial:ISBN-Suche/9783319711225" class="internal mw-magiclink-isbn">ISBN 978-3-319-71122-5</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-319-71123-2">10.1007/978-3-319-71123-2</a></span> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Hansj%C3%B6rg+Kielh%C3%B6fer&rft.btitle=Calculus+of+Variations&rft.date=2018&rft.doi=10.1007%2F978-3-319-71123-2&rft.genre=book&rft.isbn=9783319711225&rft.place=Cham&rft.pub=Springer+International+Publishing&rft.series=Texts+in+Applied+Mathematics&rft.volume=67" style="display:none"> </span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a></span> <span class="reference-text">Peter Steinke: <cite style="font-style:italic">Einleitung</cite>. In: <cite style="font-style:italic">Finite-Elemente-Methode: Rechnergestützte Einführung</cite>. Springer, Berlin/Heidelberg 2004, <a href="/wiki/Spezial:ISBN-Suche/9783662072400" class="internal mw-magiclink-isbn">ISBN 978-3-662-07240-0</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>1–11</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-662-07240-0_1">10.1007/978-3-662-07240-0_1</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=Einleitung&rft.au=Peter+Steinke&rft.btitle=Finite-Elemente-Methode%3A+Rechnergest%C3%BCtzte+Einf%C3%BChrung&rft.date=2004&rft.doi=10.1007%2F978-3-662-07240-0_1&rft.genre=book&rft.isbn=9783662072400&rft.pages=1-11&rft.place=Berlin%2FHeidelberg&rft.pub=Springer" style="display:none"> </span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text">G. Sardanashvily: <cite style="font-style:italic">Classical field theory. Advanced mathematical formulation</cite>. In: <cite style="font-style:italic">International Journal of Geometric Methods in Modern Physics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>05</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>07</span>, November 2008, <a href="/wiki/Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220219-8878%22&key=cql">0219-8878</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>1163–1189</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1142/S0219887808003247">10.1142/S0219887808003247</a></span>, <a href="/wiki/ArXiv" title="ArXiv">arxiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0811.0331">0811.0331 [abs]</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=Classical+field+theory.+Advanced+mathematical+formulation&rft.au=G.+Sardanashvily&rft.date=2008-11&rft.doi=10.1142%2FS0219887808003247&rft.genre=journal&rft.issn=0219-8878&rft.issue=07&rft.jtitle=International+Journal+of+Geometric+Methods+in+Modern+Physics&rft.pages=1163-1189&rft.volume=05" style="display:none"> </span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a></span> <span class="reference-text"><a href="/wiki/Michael_Struwe" title="Michael Struwe">Michael Struwe</a>: <cite style="font-style:italic">Plateau’s Problem and the Calculus of Variations. (MN-35)</cite>. Princeton University Press, 1989, <a href="/wiki/Spezial:ISBN-Suche/9781400860210" class="internal mw-magiclink-isbn">ISBN 978-1-4008-6021-0</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1515/9781400860210">10.1515/9781400860210</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Michael+Struwe&rft.btitle=Plateau%E2%80%99s+Problem+and+the+Calculus+of+Variations.+%28MN-35%29&rft.date=1989-12-31&rft.doi=10.1515%2F9781400860210&rft.genre=book&rft.isbn=9781400860210&rft.pub=Princeton+University+Press" style="display:none"> </span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><a href="#cite_ref-16">↑</a></span> <span class="reference-text"><a href="/wiki/Michael_Struwe" title="Michael Struwe">Michael Struwe</a>: <cite style="font-style:italic">Variational Methods</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>34</span>. Springer, Berlin/Heidelberg 2008, <a href="/wiki/Spezial:ISBN-Suche/9783540740124" class="internal mw-magiclink-isbn">ISBN 978-3-540-74012-4</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-540-74013-1">10.1007/978-3-540-74013-1</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Michael+Struwe&rft.btitle=Variational+Methods&rft.date=2008&rft.doi=10.1007%2F978-3-540-74013-1&rft.genre=book&rft.isbn=9783540740124&rft.place=Berlin%2FHeidelberg&rft.pub=Springer&rft.volume=34" style="display:none"> </span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><a href="#cite_ref-17">↑</a></span> <span class="reference-text"><a href="/wiki/J%C3%BCrgen_Jost" title="Jürgen Jost">Jürgen Jost</a>: <cite style="font-style:italic">The Dirichlet Principle. Variational Methods for the Solution of PDEs (Existence Techniques III)</cite>. In: <cite style="font-style:italic">Partial Differential Equations</cite>. Springer, New York, NY 2013, <a href="/wiki/Spezial:ISBN-Suche/9781461448099" class="internal mw-magiclink-isbn">ISBN 978-1-4614-4809-9</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>215–253</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-1-4614-4809-9_10">10.1007/978-1-4614-4809-9_10</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=The+Dirichlet+Principle.+Variational+Methods+for+the+Solution+of+PDEs+%28Existence+Techniques+III%29&rft.au=J%C3%BCrgen+Jost&rft.btitle=Partial+Differential+Equations&rft.date=2013&rft.doi=10.1007%2F978-1-4614-4809-9_10&rft.genre=book&rft.isbn=9781461448099&rft.pages=215-253&rft.place=New+York%2C+NY&rft.pub=Springer" style="display:none"> </span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><a href="#cite_ref-18">↑</a></span> <span class="reference-text">Kunio Yasue: <cite style="font-style:italic">Stochastic calculus of variations</cite>. In: <cite style="font-style:italic">Journal of Functional Analysis</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>41</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>3</span>, 1. Mai 1981, <a href="/wiki/Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220022-1236%22&key=cql">0022-1236</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>327–340</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1016/0022-1236%2881%2990079-3">10.1016/0022-1236(81)90079-3</a></span> (<a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/0022123681900793">sciencedirect.com</a> [abgerufen am 17. Oktober 2022]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=Stochastic+calculus+of+variations&rft.au=Kunio+Yasue&rft.date=1981-05-01&rft.doi=10.1016%2F0022-1236%2881%2990079-3&rft.genre=journal&rft.issn=0022-1236&rft.issue=3&rft.jtitle=Journal+of+Functional+Analysis&rft.pages=327-340&rft.volume=41" style="display:none"> </span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><a href="#cite_ref-19">↑</a></span> <span class="reference-text"><a href="/wiki/Wolfgang_Yourgrau" title="Wolfgang Yourgrau">Wolfgang Yourgrau</a>, <a href="/wiki/Stanley_Mandelstam" title="Stanley Mandelstam">Stanley Mandelstam</a>: <cite style="font-style:italic">Variational principles in dynamics and quantum theory</cite>. 3. Auflage. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, 1968, <a href="/wiki/Spezial:ISBN-Suche/0273402870" class="internal mw-magiclink-isbn">ISBN 0-273-40287-0</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Wolfgang+Yourgrau%2C+Stanley+Mandelstam&rft.btitle=Variational+principles+in+dynamics+and+quantum+theory&rft.date=1968&rft.edition=3&rft.genre=book&rft.isbn=0273402870&rft.pub=Dover+Publications" style="display:none"> </span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><a href="#cite_ref-20">↑</a></span> <span class="reference-text">Francis Clarke: <cite style="font-style:italic">Functional Analysis, Calculus of Variations and Optimal Control</cite> (= <cite style="font-style:italic"><a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a></cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>264</span>). Springer London, London 2013, <a href="/wiki/Spezial:ISBN-Suche/9781447148197" class="internal mw-magiclink-isbn">ISBN 978-1-4471-4819-7</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-1-4471-4820-3">10.1007/978-1-4471-4820-3</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Francis+Clarke&rft.btitle=Functional+Analysis%2C+Calculus+of+Variations+and+Optimal+Control&rft.date=2013&rft.doi=10.1007%2F978-1-4471-4820-3&rft.genre=book&rft.isbn=9781447148197&rft.place=London&rft.pub=Springer+London&rft.series=Graduate+Texts+in+Mathematics" style="display:none"> </span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><a href="#cite_ref-21">↑</a></span> <span class="reference-text"><a href="/w/index.php?title=Arnold_Dresden&action=edit&redlink=1" class="new" title="Arnold Dresden (Seite nicht vorhanden)">Arnold Dresden</a>: <cite style="font-style:italic">Book Review: Fondamenti di Calcolo delle Variazioni</cite>. In: <cite style="font-style:italic">Bulletin of the American Mathematical Society</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>32</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>4</span>, 1926, <a href="/wiki/Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220002-9904%22&key=cql">0002-9904</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>381–387</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1090/S0002-9904-1926-04231-2">10.1090/S0002-9904-1926-04231-2</a></span> (<a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1926-32-04/S0002-9904-1926-04231-2/home.html">ams.org</a> [abgerufen am 19. Oktober 2022]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=Book+Review%3A+Fondamenti+di+Calcolo+delle+Variazioni&rft.au=Arnold+Dresden&rft.date=1926&rft.doi=10.1090%2FS0002-9904-1926-04231-2&rft.genre=journal&rft.issn=0002-9904&rft.issue=4&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.pages=381-387&rft.volume=32" style="display:none"> </span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><a href="#cite_ref-22">↑</a></span> <span class="reference-text">Philippe Blanchard, Erwin Brüning: <cite style="font-style:italic">Direkte Methoden der Variationsrechnung</cite>. Springer Vienna, Vienna 1982, <a href="/wiki/Spezial:ISBN-Suche/9783709122617" class="internal mw-magiclink-isbn">ISBN 978-3-7091-2261-7</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-7091-2260-0">10.1007/978-3-7091-2260-0</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Philippe+Blanchard%2C+Erwin+Br%C3%BCning&rft.btitle=Direkte+Methoden+der+Variationsrechnung&rft.date=1982&rft.doi=10.1007%2F978-3-7091-2260-0&rft.genre=book&rft.isbn=9783709122617&rft.place=Vienna&rft.pub=Springer+Vienna" style="display:none"> </span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><a href="#cite_ref-23">↑</a></span> <span class="reference-text"><a href="/wiki/Bernard_Dacorogna" title="Bernard Dacorogna">Bernard Dacorogna</a>: <cite style="font-style:italic">Direct Methods in the Calculus of Variations</cite>. Springer New York, New York, NY 2007, <a href="/wiki/Spezial:ISBN-Suche/9780387357799" class="internal mw-magiclink-isbn">ISBN 978-0-387-35779-9</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-0-387-55249-1">10.1007/978-0-387-55249-1</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=Bernard+Dacorogna&rft.btitle=Direct+Methods+in+the+Calculus+of+Variations&rft.date=2007&rft.doi=10.1007%2F978-0-387-55249-1&rft.genre=book&rft.isbn=9780387357799&rft.place=New+York%2C+NY&rft.pub=Springer+New+York" style="display:none"> </span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><a href="#cite_ref-24">↑</a></span> <span class="reference-text"><a href="/wiki/R._Tyrrell_Rockafellar" class="mw-redirect" title="R. Tyrrell Rockafellar">R. Tyrrell Rockafellar</a>, <a href="/wiki/Roger_Wets" title="Roger Wets">Roger J. B. Wets</a>: <cite style="font-style:italic">Variational Analysis</cite> (= <cite style="font-style:italic">Grundlehren der mathematischen Wissenschaften</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>317</span>). Springer, Berlin/Heidelberg 1998, <a href="/wiki/Spezial:ISBN-Suche/9783540627722" class="internal mw-magiclink-isbn">ISBN 978-3-540-62772-2</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-642-02431-3">10.1007/978-3-642-02431-3</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.au=R.+Tyrrell+Rockafellar%2C+Roger+J.+B.+Wets&rft.btitle=Variational+Analysis&rft.date=1998&rft.doi=10.1007%2F978-3-642-02431-3&rft.genre=book&rft.isbn=9783540627722&rft.place=Berlin%2FHeidelberg&rft.pub=Springer&rft.series=Grundlehren+der+mathematischen+Wissenschaften" style="display:none"> </span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><a href="#cite_ref-25">↑</a></span> <span class="reference-text"><a href="/wiki/Wladimir_Iwanowitsch_Smirnow" title="Wladimir Iwanowitsch Smirnow">Wladimir I. Smirnow</a>: <i>Lehrgang der höheren Mathematik</i> (= <i>Hochschulbücher für Mathematik.</i> Bd. 5a). Teil 4, 1. (14. Auflage, deutschsprachige Ausgabe der 6. russischen Auflage). VEB Deutscher Verlag der Wissenschaften, Berlin 1988, <a href="/wiki/Spezial:ISBN-Suche/3326003668" class="internal mw-magiclink-isbn">ISBN 3-326-00366-8</a>.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><a href="#cite_ref-26">↑</a></span> <span class="reference-text">Siehe auch Helmut Fischer, Helmut Kaul: <i>Mathematik für Physiker.</i> Band 3: <i>Variationsrechnung, Differentialgeometrie, mathematische Grundlagen der allgemeinen Relativitätstheorie.</i> 2., überarbeitete Auflage. Teubner, Stuttgart u. a. 2006, <a href="/wiki/Spezial:ISBN-Suche/3835100319" class="internal mw-magiclink-isbn">ISBN 3-8351-0031-9</a>.</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><a href="#cite_ref-27">↑</a></span> <span class="reference-text">Sadri Hassani: <cite style="font-style:italic">Calculus of Variations, Symmetries, and Conservation Laws</cite>. In: <cite style="font-style:italic">Mathematical Physics</cite>. Springer International Publishing, Cham 2013, <a href="/wiki/Spezial:ISBN-Suche/9783319011943" class="internal mw-magiclink-isbn">ISBN 978-3-319-01194-3</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>1047–1075</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-319-01195-0_33">10.1007/978-3-319-01195-0_33</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Variationsrechnung&rft.atitle=Calculus+of+Variations%2C+Symmetries%2C+and+Conservation+Laws&rft.au=Sadri+Hassani&rft.btitle=Mathematical+Physics&rft.date=2013&rft.doi=10.1007%2F978-3-319-01195-0_33&rft.genre=book&rft.isbn=9783319011943&rft.pages=1047-1075&rft.place=Cham&rft.pub=Springer+International+Publishing" style="display:none"> </span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><a href="#cite_ref-28">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Euler_operator"><i>Euler operator – Encyclopedia of Mathematics.</i></a><span class="Abrufdatum"> Abgerufen am 21. Oktober 2022</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AVariationsrechnung&rft.title=Euler+operator+%E2%80%93+Encyclopedia+of+Mathematics&rft.description=Euler+operator+%E2%80%93+Encyclopedia+of+Mathematics&rft.identifier=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FEuler_operator"> </span></span> </li> </ol> <div class="hintergrundfarbe1 rahmenfarbe1 navigation-not-searchable normdaten-typ-s" style="border-style: solid; 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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 – Asturisch" lang="ast" hreflang="ast" data-title="Cálculu de variaciones" data-language-autonym="Asturianu" data-language-local-name="Asturisch" 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 – Baschkirisch" lang="ba" hreflang="ba" data-title="Вариациалы иҫәпләмә" data-language-autonym="Башҡортса" data-language-local-name="Baschkirisch" 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 – Belarussisch" lang="be" hreflang="be" data-title="Варыяцыйнае злічэнне" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" 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 – Bulgarisch" lang="bg" hreflang="bg" data-title="Вариационно смятане" data-language-autonym="Български" data-language-local-name="Bulgarisch" 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 – Katalanisch" lang="ca" hreflang="ca" data-title="Càlcul de variacions" data-language-autonym="Català" data-language-local-name="Katalanisch" 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 – Tschechisch" lang="cs" hreflang="cs" data-title="Variační počet" data-language-autonym="Čeština" data-language-local-name="Tschechisch" 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 – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Вариацилле шутлав" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target"><span>Чӑвашла</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 – Griechisch" lang="el" hreflang="el" data-title="Λογισμός των μεταβολών" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" 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 – Englisch" lang="en" hreflang="en" data-title="Calculus of variations" data-language-autonym="English" data-language-local-name="Englisch" 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 – Spanisch" lang="es" hreflang="es" data-title="Cálculo de variaciones" data-language-autonym="Español" data-language-local-name="Spanisch" 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 – Estnisch" lang="et" hreflang="et" data-title="Variatsioonarvutus" data-language-autonym="Eesti" data-language-local-name="Estnisch" 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 – Baskisch" lang="eu" hreflang="eu" data-title="Bariazioen kalkulu" data-language-autonym="Euskara" data-language-local-name="Baskisch" 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 – Persisch" lang="fa" hreflang="fa" data-title="حساب تغییرات" data-language-autonym="فارسی" data-language-local-name="Persisch" 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 – Finnisch" lang="fi" hreflang="fi" data-title="Variaatiolaskenta" data-language-autonym="Suomi" data-language-local-name="Finnisch" 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 – Französisch" lang="fr" hreflang="fr" data-title="Calcul des variations" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/C%C3%A1lculo_de_variaci%C3%B3ns" title="Cálculo de variacións – Galicisch" lang="gl" hreflang="gl" data-title="Cálculo de variacións" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</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 – Hebräisch" lang="he" hreflang="he" data-title="חשבון וריאציות" data-language-autonym="עברית" data-language-local-name="Hebräisch" 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 – Ungarisch" lang="hu" hreflang="hu" data-title="Variációszámítás" data-language-autonym="Magyar" data-language-local-name="Ungarisch" 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 – Italienisch" lang="it" hreflang="it" data-title="Calcolo delle variazioni" data-language-autonym="Italiano" data-language-local-name="Italienisch" 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 – Japanisch" lang="ja" hreflang="ja" data-title="変分法" data-language-autonym="日本語" data-language-local-name="Japanisch" 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 – Kasachisch" lang="kk" hreflang="kk" data-title="Вариациялық есептеу" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" 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 – Koreanisch" lang="ko" hreflang="ko" data-title="변분법" data-language-autonym="한국어" data-language-local-name="Koreanisch" 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 – Maltesisch" lang="mt" hreflang="mt" data-title="Kalkulu tal-varjazzjonijiet" data-language-autonym="Malti" data-language-local-name="Maltesisch" 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 – Niederländisch" lang="nl" hreflang="nl" data-title="Variatierekening" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" 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 – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Variasjonsrekning" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (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 – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Variasjonsregning" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (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 – Polnisch" lang="pl" hreflang="pl" data-title="Rachunek wariacyjny" data-language-autonym="Polski" data-language-local-name="Polnisch" 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 – Piemontesisch" lang="pms" hreflang="pms" data-title="Càlcol dle variassion" data-language-autonym="Piemontèis" data-language-local-name="Piemontesisch" 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 – Portugiesisch" lang="pt" hreflang="pt" data-title="Cálculo variacional" data-language-autonym="Português" data-language-local-name="Portugiesisch" 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 – Rumänisch" lang="ro" hreflang="ro" data-title="Calcul variațional" data-language-autonym="Română" data-language-local-name="Rumänisch" 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 – Russisch" lang="ru" hreflang="ru" data-title="Вариационное исчисление" data-language-autonym="Русский" data-language-local-name="Russisch" 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 – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Calculus of variations" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" 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 – Slowakisch" lang="sk" hreflang="sk" data-title="Variačný počet" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" 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 – Slowenisch" lang="sl" hreflang="sl" data-title="Variacijski račun" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" 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 – Albanisch" lang="sq" hreflang="sq" data-title="Analiza e variacionit" data-language-autonym="Shqip" data-language-local-name="Albanisch" 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 – Serbisch" lang="sr" hreflang="sr" data-title="Varijacijski račun" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" 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 – Schwedisch" lang="sv" hreflang="sv" data-title="Variationskalkyl" data-language-autonym="Svenska" data-language-local-name="Schwedisch" 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 – Ukrainisch" lang="uk" hreflang="uk" data-title="Варіаційне числення" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_t%C3%ADnh_bi%E1%BA%BFn_ph%C3%A2n" title="Phép tính biến phân – Vietnamesisch" lang="vi" hreflang="vi" data-title="Phép tính biến phân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8F%98%E5%88%86%E6%B3%95" title="变分法 – Wu" lang="wuu" hreflang="wuu" data-title="变分法" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%98%E5%88%86%E6%B3%95" title="变分法 – Chinesisch" lang="zh" hreflang="zh" data-title="变分法" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%AE%8A%E5%88%86%E6%B3%95" title="變分法 – Kantonesisch" lang="yue" hreflang="yue" data-title="變分法" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q216861#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" 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