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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Lagrange-Dichte</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>Lagrange-Dichte</b> <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 {\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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9027196ecb178d598958555ea01c43157d83597c.svg" 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}}}" loading="lazy"></span> (nach dem Mathematiker <a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a>) spielt in der <a href="Theoretische_Physik" class="mw-redirect" title="Theoretische Physik">theoretischen Physik</a> eine Rolle bei der Betrachtung von <a href="Feld_(Physik)" title="Feld (Physik)">Feldern</a>. Sie beschreibt die Dichte der <a href="Lagrange-Funktion" class="mw-redirect" title="Lagrange-Funktion">Lagrange-Funktion</a> <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 L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> in einem Volumenelement. Daher ist die Lagrange-Funktion definiert als das <a href="Integralrechnung" title="Integralrechnung">Integral</a> der Lagrange-Dichte über dem betrachteten Volumen:
</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 L=\int \mathrm {d} ^{3}r{\mathcal {L}}=\iiint \mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z\,{\mathcal {L}}\left(\phi ,{\frac {\partial \phi }{\partial t}},{\frac {\partial \phi }{\partial x}},{\frac {\partial \phi }{\partial y}},{\frac {\partial \phi }{\partial z}},t\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mo>∫<!-- ∫ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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</msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
<mo>=</mo>
<mo>∭<!-- ∭ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>y</mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
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</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>ϕ<!-- ϕ --></mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>z</mi>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L=\int \mathrm {d} ^{3}r{\mathcal {L}}=\iiint \mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z\,{\mathcal {L}}\left(\phi ,{\frac {\partial \phi }{\partial t}},{\frac {\partial \phi }{\partial x}},{\frac {\partial \phi }{\partial y}},{\frac {\partial \phi }{\partial z}},t\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20307cc3019b1e6694ec33c2c592282a8f2981ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.924ex; height:6.176ex;" alt="{\displaystyle L=\int \mathrm {d} ^{3}r{\mathcal {L}}=\iiint \mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z\,{\mathcal {L}}\left(\phi ,{\frac {\partial \phi }{\partial t}},{\frac {\partial \phi }{\partial x}},{\frac {\partial \phi }{\partial y}},{\frac {\partial \phi }{\partial z}},t\right)}" loading="lazy"></span></dd></dl>
<p>mit dem betrachteten Feld <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 \phi (x,y,z,t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi (x,y,z,t)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b4b2ff559652e6fa03c0e20e596711bb8740ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.709ex; height:2.843ex;" alt="{\displaystyle \phi (x,y,z,t)}" loading="lazy"></span>.
</p><p>Der eigentliche Zweck der Lagrange-Dichte ist die Beschreibung von Feldern durch <a href="Bewegungsgleichung" title="Bewegungsgleichung">Bewegungsgleichungen</a>. So, wie man die <a href="Lagrange-Funktion" class="mw-redirect" title="Lagrange-Funktion">Lagrange-Gleichungen</a> zweiter Art aus dem Hamiltonschen Prinzip erhält, kann man die Lagrange-Gleichungen für Felder aus dem <a href="Hamiltonsches_Prinzip#Das_Hamiltonsche_Prinzip_f.C3.BCr_Felder" title="Hamiltonsches Prinzip">Hamiltonschen Prinzip für Felder</a> erhalten (<a href="Feldtheorie_(Physik)#Die_Bewegungsgleichung_für_Felder" title="Feldtheorie (Physik)">Herleitung</a>). Entsprechend lautet die Bewegungsgleichung:
</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 {\frac {\partial {\mathcal {L}}}{\partial \phi _{i}}}-{\frac {\partial }{\partial t}}{\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi _{i}}{\partial t}}}}-\sum _{j=1}^{3}{\frac {\partial }{\partial x_{j}}}{\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi _{i}}{\partial x_{j}}}}}={\frac {\partial {\mathcal {L}}}{\partial \phi _{i}}}-\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{i})}}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<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>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<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>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
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</msub>
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</mfrac>
</mrow>
<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>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>=</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>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msub>
<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>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msub>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \phi _{i}}}-{\frac {\partial }{\partial t}}{\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi _{i}}{\partial t}}}}-\sum _{j=1}^{3}{\frac {\partial }{\partial x_{j}}}{\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi _{i}}{\partial x_{j}}}}}={\frac {\partial {\mathcal {L}}}{\partial \phi _{i}}}-\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{i})}}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab26e2b48c8d04d52308503cb8911119424cfcd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:57.857ex; height:8.676ex;" alt="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \phi _{i}}}-{\frac {\partial }{\partial t}}{\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi _{i}}{\partial t}}}}-\sum _{j=1}^{3}{\frac {\partial }{\partial x_{j}}}{\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi _{i}}{\partial x_{j}}}}}={\frac {\partial {\mathcal {L}}}{\partial \phi _{i}}}-\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{i})}}=0}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<p>Für eine in einer Dimension <a href="Schwingung" title="Schwingung">schwingende</a> <a href="Saite" title="Saite">Saite</a> ergibt sich für die <b>Lagrange-Dichte</b>
</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 {\mathcal {L}}={\frac {1}{2}}\left[\mu \left({\frac {\partial \phi }{\partial t}}\right)^{2}-E\left({\frac {\partial \phi }{\partial x}}\right)^{2}\right]}">
<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>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<mi>μ<!-- μ --></mi>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mi>E</mi>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>]</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left[\mu \left({\frac {\partial \phi }{\partial t}}\right)^{2}-E\left({\frac {\partial \phi }{\partial x}}\right)^{2}\right]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/325955baaad2289b046f6c97bedc7364c9b1d14d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.846ex; height:7.509ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left[\mu \left({\frac {\partial \phi }{\partial t}}\right)^{2}-E\left({\frac {\partial \phi }{\partial x}}\right)^{2}\right]}" loading="lazy"></span></dd></dl>
<p>In diesem Beispiel bedeuten:
</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 \phi =\phi (x,t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo>=</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi =\phi (x,t)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d5de886eb0ea568d064f5ca1ee40b8d936f784.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.882ex; height:2.843ex;" alt="{\displaystyle \phi =\phi (x,t)}" loading="lazy"></span> die Auslenkung eines Punktes der Saite aus der Ruhelage (Feldvariable)</dd>
<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 \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> die lineare Massendichte</dd>
<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 E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> den <a href="Elastizit%C3%A4tsmodul" title="Elastizitätsmodul">Elastizitätsmodul</a></dd></dl>
<p>Mit dieser <b>Lagrange-Dichte</b> ergibt sich
</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 {\frac {\partial {\mathcal {L}}}{\partial \phi }}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
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<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>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \phi }}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c68071933349d965f1db1cbe58822342a03d40c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.019ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \phi }}=0}" loading="lazy"></span></dd></dl>
<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 {\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi }{\partial t}}}}=\mu {\frac {\partial \phi }{\partial t}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<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>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi }{\partial t}}}}=\mu {\frac {\partial \phi }{\partial t}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1861ad09681fd3a64c4923d92eee643e104fb736.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:12.942ex; height:7.676ex;" alt="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi }{\partial t}}}}=\mu {\frac {\partial \phi }{\partial t}}}" loading="lazy"></span></dd></dl>
<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 {\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi }{\partial x}}}}=-E{\frac {\partial \phi }{\partial x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<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>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi }{\partial x}}}}=-E{\frac {\partial \phi }{\partial x}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb94ef77d7bb315a377a74e8acaebd73e0e390b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:15.124ex; height:7.676ex;" alt="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi }{\partial x}}}}=-E{\frac {\partial \phi }{\partial x}}}" loading="lazy"></span></dd></dl>
<p>Damit ergibt sich für die <b>Bewegungsgleichung</b> der schwingenden Saite
</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 E{\frac {\partial ^{2}\phi }{\partial x^{2}}}-\mu {\frac {\partial ^{2}\phi }{\partial t^{2}}}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E{\frac {\partial ^{2}\phi }{\partial x^{2}}}-\mu {\frac {\partial ^{2}\phi }{\partial t^{2}}}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15f8d74cc2db23498edb0a5f7b34aa91327ad698.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.517ex; height:6.009ex;" alt="{\displaystyle E{\frac {\partial ^{2}\phi }{\partial x^{2}}}-\mu {\frac {\partial ^{2}\phi }{\partial t^{2}}}=0}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Anwendung_in_der_Relativitätstheorie"><span id="Anwendung_in_der_Relativit.C3.A4tstheorie"></span>Anwendung in der Relativitätstheorie</h2></div>
<p>Anwendung findet die Beschreibung physikalischer Vorgänge über die Lagrange-Dichte statt über die Lagrange-Funktion vor allem in relativistischen Vorgängen. Hier ist eine <a href="Kovarianz_(Physik)" title="Kovarianz (Physik)">kovariante Darstellung</a> der Lagrange-Funktion gewünscht, dann ist die <a href="Wirkung_(Physik)" title="Wirkung (Physik)">Wirkung</a> über
</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 S=\int \mathrm {d} ^{4}x\,{\sqrt {-g}}\,{\mathcal {L}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>=</mo>
<mo>∫<!-- ∫ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mo>−<!-- − --></mo>
<mi>g</mi>
</msqrt>
</mrow>
<mspace width="thinmathspace"></mspace>
<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 S=\int \mathrm {d} ^{4}x\,{\sqrt {-g}}\,{\mathcal {L}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d82b0c92862971e6e49761426570377be07a73d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.481ex; height:5.676ex;" alt="{\displaystyle S=\int \mathrm {d} ^{4}x\,{\sqrt {-g}}\,{\mathcal {L}}}" loading="lazy"></span></dd></dl>
<p>definiert, wobei <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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> die <a href="Determinante" title="Determinante">Determinante</a> des <a href="Metrischer_Tensor" title="Metrischer Tensor">metrischen Tensors</a> ist.<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> Damit ist die Lagrange-Funktion ein Lorentz-Pseudoskalar, also invariant unter <a href="Lorentz-Transformation" title="Lorentz-Transformation">Lorentz-Transformationen</a>:
</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 {\mathcal {L}}'(x_{\mu })={\mathcal {L}}(x'_{\mu })={\mathcal {L}}(x_{\mu })}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
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<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
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<mo stretchy="false">(</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
<mo>′</mo>
</msubsup>
<mo stretchy="false">)</mo>
<mo>=</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
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<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}'(x_{\mu })={\mathcal {L}}(x'_{\mu })={\mathcal {L}}(x_{\mu })}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46063fb5b6c0a330b8bd93c7198cf4d63cec0820.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.78ex; height:3.176ex;" alt="{\displaystyle {\mathcal {L}}'(x_{\mu })={\mathcal {L}}(x'_{\mu })={\mathcal {L}}(x_{\mu })}" loading="lazy"></span> 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 x'_{\mu }=\Lambda _{\mu \nu }x^{\nu }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
<mo>′</mo>
</msubsup>
<mo>=</mo>
<msub>
<mi mathvariant="normal">Λ<!-- Λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x'_{\mu }=\Lambda _{\mu \nu }x^{\nu }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/701c6a3a922dc008ec1199606738db4b34252ae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.792ex; height:3.009ex;" alt="{\displaystyle x'_{\mu }=\Lambda _{\mu \nu }x^{\nu }}" loading="lazy"></span>, wobei <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 \Lambda _{\mu \nu }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Λ<!-- Λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Lambda _{\mu \nu }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad710608fd9eee79f285bc92ff78f1348611e876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.708ex; height:2.843ex;" alt="{\displaystyle \Lambda _{\mu \nu }}" loading="lazy"></span> der Lorentz-Transformationstensor ist.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Franz_Schwabl" title="Franz Schwabl">Franz Schwabl</a>: <i>Lagrange-Dichte</i>. In: Ders.: <i>Quantenmechanik für Fortgeschrittene (QM II)</i>. Springer, Berlin 2005, ISBN 978-3-540-28865-7, S. 281ff.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></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">Clinton L. Lewis: <cite style="font-style:italic">Explicit gauge covariant Euler–Lagrange equation</cite>. In: <cite style="font-style:italic">American Journal of Physics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>77</span>, 2009, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>839</span>, <a href="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/10.1119/1.3153503">10.1119/1.3153503</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:Lagrange-Dichte&rft.atitle=Explicit+gauge+covariant+Euler-Lagrange+equation&rft.au=Clinton+L.+Lewis&rft.btitle=American+Journal+of+Physics&rft.date=2009&rft.doi=10.1119%2F1.3153503&rft.genre=book&rft.pages=839&rft.volume=77" style="display:none"> </span></span>
</li>
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