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<title>Erste Variation</title>
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<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Erste_Variation rootpage-Erste_Variation skin-vector-2022 action-view">
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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Erste Variation</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>erste Variation</b> ist eine verallgemeinerte <a href="Richtungsableitung" title="Richtungsableitung">Richtungsableitung</a> eines <a href="Funktional" title="Funktional">Funktionals</a>.
Ihre Eigenschaften sind in der angewandten <a href="Mathematik" title="Mathematik">Mathematik</a> und der <a href="Theoretische_Physik" class="mw-redirect" title="Theoretische Physik">theoretischen Physik</a> relevant.
Die erste Variation spielt eine zentrale Rolle in der <a href="Variationsrechnung" title="Variationsrechnung">Variationsrechnung</a> und wird in der <a href="Analytische_Mechanik" class="mw-redirect" title="Analytische Mechanik">analytischen Mechanik</a> genutzt.
Ein verwandtes Konzept ist die <a href="Funktionalableitung" title="Funktionalableitung">Funktionalableitung</a>.
</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Sei <span class="mwe-math-element mwe-math-element-inline"><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy"></span> ein Funktionenraum; <span class="mwe-math-element mwe-math-element-inline"><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\rightarrow \mathbb {K} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">K</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J:X\rightarrow \mathbb {K} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/209d001872ebd08d473c285bf5af5e0a1409f3ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.811ex; height:2.176ex;" alt="{\displaystyle J:X\rightarrow \mathbb {K} }" loading="lazy"></span> ein Funktional mit <span class="mwe-math-element mwe-math-element-inline"><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 {K} =\mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">K</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {K} =\mathbb {R} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaa0cac96c9853f0cba61605679cf8963983b5d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.585ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} =\mathbb {R} }" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><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 {K} =\mathbb {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">K</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {K} =\mathbb {C} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a1a8ee77fce00b8ddbd288a297dccae8eab7f8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.585ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} =\mathbb {C} }" loading="lazy"></span>; <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y,h\in X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>,</mo>
<mi>h</mi>
<mo>∈<!-- ∈ --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y,h\in X}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ff6e016bf5fe9681e72405d3ed2af472e975307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.349ex; height:2.509ex;" alt="{\displaystyle y,h\in X}" loading="lazy"></span> Funktionen und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon \in \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ε<!-- ε --></mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varepsilon \in \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7dd62e0a7627b820957f2c201ab6e4f9d8186135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle \varepsilon \in \mathbb {R} }" loading="lazy"></span>.
Dann ist die <b>erste Variation</b> des Funktionals <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span> nach <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> definiert als
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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 J(y)(h)=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>δ<!-- δ --></mi>
<mi>J</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<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>ε<!-- ε --></mi>
</mrow>
</mfrac>
</mrow>
<mi>J</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>+</mo>
<mi>ε<!-- ε --></mi>
<mi>h</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>ε<!-- ε --></mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta J(y)(h)=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0add8563ab857f6d034d6133da7c8d70bdb3c93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.388ex; height:5.843ex;" alt="{\displaystyle \delta J(y)(h)=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0}}" loading="lazy"></span>.</dd></dl>
<p>Dies entspricht dem <a href="G%C3%A2teaux-Differential" title="Gâteaux-Differential">Gâteaux-Differential</a> des Funktionals <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span> an der Stelle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> in Richtung <span class="mwe-math-element mwe-math-element-inline"><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" 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}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<ol><li>Die erste Variation ist eine <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildung</a>:
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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 (F(y)+\alpha G(y))(h)=\delta F(y)(h)+\alpha \delta G(y)(h)\quad \forall \alpha \in \mathbb {K} ,\,\forall F,G\in {\mathcal {D}}^{\prime }(\Omega )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>δ<!-- δ --></mi>
<mo stretchy="false">(</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>α<!-- α --></mi>
<mi>G</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>δ<!-- δ --></mi>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>α<!-- α --></mi>
<mi>δ<!-- δ --></mi>
<mi>G</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mspace width="1em"></mspace>
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>α<!-- α --></mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">K</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>F</mi>
<mo>,</mo>
<mi>G</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">D</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta (F(y)+\alpha G(y))(h)=\delta F(y)(h)+\alpha \delta G(y)(h)\quad \forall \alpha \in \mathbb {K} ,\,\forall F,G\in {\mathcal {D}}^{\prime }(\Omega )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b81dbac745166db0e1477e7ab9fedb8a881dad22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.02ex; height:3.009ex;" alt="{\displaystyle \delta (F(y)+\alpha G(y))(h)=\delta F(y)(h)+\alpha \delta G(y)(h)\quad \forall \alpha \in \mathbb {K} ,\,\forall F,G\in {\mathcal {D}}^{\prime }(\Omega )}" loading="lazy"></span></dd></dl></li>
<li>Für ein Produkt aus Funktionalen <span class="mwe-math-element mwe-math-element-inline"><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(y)=G(y)H(y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>G</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mi>H</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(y)=G(y)H(y)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3038f6ba477ea2721583fd2e786def7e5d6947ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.624ex; height:2.843ex;" alt="{\displaystyle F(y)=G(y)H(y)}" loading="lazy"></span> gilt die Produktregel:
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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 F(y)(h)=\delta G(y)(h)\ H(y)+G(y)\ \delta H(y)(h)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>δ<!-- δ --></mi>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>δ<!-- δ --></mi>
<mi>G</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mtext>&nbsp;</mtext>
<mi>H</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>G</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
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<mi>δ<!-- δ --></mi>
<mi>H</mi>
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<annotation encoding="application/x-tex">{\displaystyle \delta F(y)(h)=\delta G(y)(h)\ H(y)+G(y)\ \delta H(y)(h)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/328ebcadf396e9156d2cc5494732f0c8f1736b46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.036ex; height:2.843ex;" alt="{\displaystyle \delta F(y)(h)=\delta G(y)(h)\ H(y)+G(y)\ \delta H(y)(h)}" loading="lazy"></span></dd></dl></li></ol>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<p>Die erste Variation von
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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(y)=\int _{a}^{b}yy'\,\mathrm {d} x.}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
<mo stretchy="false">(</mo>
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<mo>=</mo>
<msubsup>
<mo>∫<!-- ∫ --></mo>
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<mi>a</mi>
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<mi>b</mi>
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<mi>y</mi>
<msup>
<mi>y</mi>
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<mspace width="thinmathspace"></mspace>
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<mo>.</mo>
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<annotation encoding="application/x-tex">{\displaystyle J(y)=\int _{a}^{b}yy'\,\mathrm {d} x.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b28bc63b014e48fa9c4e642743900fcf5bda61d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.98ex; height:6.343ex;" alt="{\displaystyle J(y)=\int _{a}^{b}yy'\,\mathrm {d} x.}" loading="lazy"></span></dd></dl>
<p>ist nach obiger Definition
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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}\delta J(y)(h)&amp;=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0}\\&amp;={\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}(y+\varepsilon h)(y^{\prime }+\varepsilon h^{\prime })\,\mathrm {d} x{\Bigg |}_{\varepsilon =0}\\&amp;={\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\,\mathrm {d} x{\Bigg |}_{\varepsilon =0}\\&amp;=\int _{a}^{b}\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\right|_{\varepsilon =0}\,\mathrm {d} x\\&amp;=\int _{a}^{b}(yh^{\prime }+y^{\prime }h+2\varepsilon hh^{\prime }){\bigg |}_{\varepsilon =0}\,\mathrm {d} x\\&amp;=\int _{a}^{b}(yh^{\prime }+y^{\prime }h)\,\mathrm {d} x\,.\end{aligned}}}">
<semantics>
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<mi>ε<!-- ε --></mi>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\delta J(y)(h)&amp;=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0}\\&amp;={\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}(y+\varepsilon h)(y^{\prime }+\varepsilon h^{\prime })\,\mathrm {d} x{\Bigg |}_{\varepsilon =0}\\&amp;={\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\,\mathrm {d} x{\Bigg |}_{\varepsilon =0}\\&amp;=\int _{a}^{b}\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\right|_{\varepsilon =0}\,\mathrm {d} x\\&amp;=\int _{a}^{b}(yh^{\prime }+y^{\prime }h+2\varepsilon hh^{\prime }){\bigg |}_{\varepsilon =0}\,\mathrm {d} x\\&amp;=\int _{a}^{b}(yh^{\prime }+y^{\prime }h)\,\mathrm {d} x\,.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d23ec73b78b8a1a3016837e563e49b2df3e4d97d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.838ex; width:53.601ex; height:38.843ex;" alt="{\displaystyle {\begin{aligned}\delta J(y)(h)&amp;=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0}\\&amp;={\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}(y+\varepsilon h)(y^{\prime }+\varepsilon h^{\prime })\,\mathrm {d} x{\Bigg |}_{\varepsilon =0}\\&amp;={\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\,\mathrm {d} x{\Bigg |}_{\varepsilon =0}\\&amp;=\int _{a}^{b}\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\right|_{\varepsilon =0}\,\mathrm {d} x\\&amp;=\int _{a}^{b}(yh^{\prime }+y^{\prime }h+2\varepsilon hh^{\prime }){\bigg |}_{\varepsilon =0}\,\mathrm {d} x\\&amp;=\int _{a}^{b}(yh^{\prime }+y^{\prime }h)\,\mathrm {d} x\,.\end{aligned}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Variation_(Mathematik)" title="Variation (Mathematik)">Variation</a></li>
<li><a href="Hamiltonsches_Prinzip" title="Hamiltonsches Prinzip">Hamiltonsches Prinzip</a></li>
<li><a href="Euler-Lagrange-Gleichung" class="mw-redirect" title="Euler-Lagrange-Gleichung">Euler-Lagrange-Gleichung</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20190103151857/http://exampleproblems.com:80/wiki/index.php/Calculus_of_Variations">Exampleproblems.com</a> hat weitere Beispiele.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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