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<title>Rayleighsche Dissipationsfunktion</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Rayleighsche Dissipationsfunktion</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">Die Rayleighsche Dissipationsfunktion ist ein von <a href="John_Strutt%2C_3._Baron_Rayleigh" title="John Strutt, 3. Baron Rayleigh">Lord Rayleigh</a> 1876<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a> eingeführter Ansatz für eine geschwindigkeitsabhängige <a href="Reibungskraft" class="mw-redirect" title="Reibungskraft">Reibungskraft</a> in der <a href="Klassische_Mechanik" title="Klassische Mechanik">klassischen Mechanik</a>. Er lässt sich auch im <a href="Lagrange-Formalismus" title="Lagrange-Formalismus">Lagrange-Formalismus</a> der klassischen Mechanik formulieren.

Der Lagrangeformalismus beschreibt die Dynamik eines Systems über die Lagrangefunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=T-V}">

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<annotation encoding="application/x-tex">{\displaystyle L=T-V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00644a2f8d4ddaa3b1ae36d472aabaeb63e4d9dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.945ex; height:2.343ex;" alt="{\displaystyle L=T-V}" loading="lazy"> (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> der kinetischen Energie und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> der potentiellen Energie), wobei diese als Funktion von generalisierten Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle q_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2752dcbff884354069fe332b8e51eb0a70a531b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.837ex; height:2.009ex;" alt="{\displaystyle q_{i}}" loading="lazy"> (und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {q_{i}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\dot {q_{i}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/326955fb5b20bfb832bdd050f4081176833bc2a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.837ex; height:2.509ex;" alt="{\displaystyle {\dot {q_{i}}}}" loading="lazy"> für die zugehörige generalisierte Geschwindigkeit) aufgefasst wird (wobei der Index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> die Komponenten bezeichnet). Dann kann man geschwindigkeitsabhängige Reibungskräfte über eine nicht-konservative generalisierte Kraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{i}^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{i}^{*}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/241cba8306330f69e5596efb2147452720868c8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.893ex; height:2.843ex;" alt="{\displaystyle Q_{i}^{*}}" loading="lazy"> auf der rechten Seite der Lagrangegleichung berücksichtigen (siehe auch den Artikel <a href="Lagrange-Formalismus" title="Lagrange-Formalismus">Lagrange-Formalismus</a>):

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q_{i}}}}}-{\frac {\partial L}{\partial q_{i}}}={\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q_{i}}}}}+{\frac {\partial T}{\partial q_{i}}}-{\frac {\partial V}{\partial q_{i}}}=Q_{i}^{*}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76ed702d6ea8918449d5711205727a7c44ec9930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.349ex; height:5.843ex;" alt="{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q_{i}}}}}-{\frac {\partial L}{\partial q_{i}}}={\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {q_{i}}}}}+{\frac {\partial T}{\partial q_{i}}}-{\frac {\partial V}{\partial q_{i}}}=Q_{i}^{*}}" loading="lazy"></dd></dl>

Rayleigh machte nun für die Reibungskraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18f9a5ed62f131c8ad9acdf7f8539f103f1e0ec9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.294ex; height:2.509ex;" alt="{\displaystyle F_{i}}" loading="lazy"> in euklidischen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}" loading="lazy"> (mit zugehöriger euklidischer Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7dffe5726650f6daac54829972a94f38eb8ec127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}" loading="lazy">) folgenden Ansatz:

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<annotation encoding="application/x-tex">{\displaystyle F_{i}=-\sum _{j}\,K_{ij}\,v_{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb2cbfc1b7ee55a555801c7c496dbf6ecc6588cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.592ex; height:5.843ex;" alt="{\displaystyle F_{i}=-\sum _{j}\,K_{ij}\,v_{j}}" loading="lazy"></dd></dl>

mit der Dissipationsmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{ij}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72478832c6c454ecf9c05248fdcc312d551f4f4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.45ex; height:2.843ex;" alt="{\displaystyle K_{ij}}" loading="lazy">. Die zugehörige Dissipationsfunktion

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<annotation encoding="application/x-tex">{\displaystyle K={\frac {1}{2}}\,\sum _{i,j}\,K_{ij}\,v_{i}\,v_{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0b09e0018502c4b7802ee3d3d7d235be2c40364.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.255ex; height:6.676ex;" alt="{\displaystyle K={\frac {1}{2}}\,\sum _{i,j}\,K_{ij}\,v_{i}\,v_{j}}" loading="lazy"></dd></dl>

ist im einfachsten Fall einer diagonalen Dissipationsmatrix<a href="#cite_note-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle K={\frac {1}{2}}\,\sum _{i}\,k_{i}\,v_{i}^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/044f9b30288913139fd4e9cc562ec0b4e85effdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.646ex; height:6.343ex;" alt="{\displaystyle K={\frac {1}{2}}\,\sum _{i}\,k_{i}\,v_{i}^{2}}" loading="lazy"></dd></dl>

Mit der Dissipationsfunktion ist die Reibungskraft demnach:

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<annotation encoding="application/x-tex">{\displaystyle F_{i}=-{\frac {\partial }{\partial v_{i}}}K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95e06af54bdd0e275893132e789e18eae45a408b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.348ex; height:5.843ex;" alt="{\displaystyle F_{i}=-{\frac {\partial }{\partial v_{i}}}K}" loading="lazy"></dd></dl>

Beim Übergang zu generalisierten Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{i}^{*}=\sum _{j}\,F_{j}\,{\frac {\partial r_{j}}{\partial q_{i}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a29e155366cdfcbd1149eb7f06cbdeef716ef02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.024ex; height:7.176ex;" alt="{\displaystyle Q_{i}^{*}=\sum _{j}\,F_{j}\,{\frac {\partial r_{j}}{\partial q_{i}}}}" loading="lazy"></dd></dl>

Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}(t,q,{\dot {q}})=\sum _{k}{\frac {\partial r_{i}}{\partial q_{k}}}\,{\dot {q_{k}}}+{\frac {\partial r_{i}}{\partial t}}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{i}(t,q,{\dot {q}})=\sum _{k}{\frac {\partial r_{i}}{\partial q_{k}}}\,{\dot {q_{k}}}+{\frac {\partial r_{i}}{\partial t}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c60f8b96b632adbfb8ad7e599480cca8ad9d1174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.567ex; height:6.509ex;" alt="{\displaystyle v_{i}(t,q,{\dot {q}})=\sum _{k}{\frac {\partial r_{i}}{\partial q_{k}}}\,{\dot {q_{k}}}+{\frac {\partial r_{i}}{\partial t}}}" loading="lazy"> gilt:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial v_{i}}{\partial {\dot {q_{k}}}}}={\frac {\partial r_{i}}{\partial q_{k}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bdbe47ba2703587282c9cc76cdc2d72535c8cf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.658ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial v_{i}}{\partial {\dot {q_{k}}}}}={\frac {\partial r_{i}}{\partial q_{k}}}}" loading="lazy"></dd></dl>

und damit

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<annotation encoding="application/x-tex">{\displaystyle Q_{i}^{*}=-\sum _{j}\,{\frac {\partial K}{\partial v_{j}}}\,{\frac {\partial v_{j}}{\partial {\dot {q_{i}}}}}=-{\frac {\partial K}{\partial {\dot {q_{i}}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba617b08f910ab1e303ac09ccf285886140d414c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.241ex; height:7.176ex;" alt="{\displaystyle Q_{i}^{*}=-\sum _{j}\,{\frac {\partial K}{\partial v_{j}}}\,{\frac {\partial v_{j}}{\partial {\dot {q_{i}}}}}=-{\frac {\partial K}{\partial {\dot {q_{i}}}}}}" loading="lazy"></dd></dl>

Daneben kann es auch andere, nicht durch einen Rayleigh-Ansatz beschreibbare nicht-konservative generalisierte Kräfte zur Beschreibung von Reibung geben.

Gegeben die Rayleighsche Dissipationsfunktion

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}={\frac {1}{2}}\sum _{i,j=1}^{n}k_{ij}{\dot {q}}_{i}{\dot {q}}_{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5632042716791bb4113cf5550f94daefd704c99a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.753ex; height:7.176ex;" alt="{\displaystyle {\mathcal {F}}={\frac {1}{2}}\sum _{i,j=1}^{n}k_{ij}{\dot {q}}_{i}{\dot {q}}_{j}}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> und Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{ij}\in \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle k_{ij}\in \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46543ac3784bb4192a9c8f4841a7012496322642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle k_{ij}\in \mathbb {R} }" loading="lazy"> lässt sich für

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<annotation encoding="application/x-tex">{\displaystyle T=T({\dot {q}}_{i},q_{i})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9839e6b50313194a5ed27eca8521d10daeb30cb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.228ex; height:2.843ex;" alt="{\displaystyle T=T({\dot {q}}_{i},q_{i})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=V(q_{i})}">

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<annotation encoding="application/x-tex">{\displaystyle V=V(q_{i})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4c71529ef1679026667a17ad5d5c237d6b73f92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.319ex; height:2.843ex;" alt="{\displaystyle V=V(q_{i})}" loading="lazy"></dd></dl>

die zeitliche <a href="%C3%84nderungsrate" title="Änderungsrate">Änderungsrate</a> der Gesamtenergie durch Reibung wie folgt angeben:<a href="#cite_note-4">[4]</a>

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</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} (T+V)}{\mathrm {d} t}}=-2{\mathcal {F}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add925dca60a8033a231e7573e79efb2087381ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.197ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} (T+V)}{\mathrm {d} t}}=-2{\mathcal {F}}}" loading="lazy"></dd></dl>

In Ausdrücken von gegen Reibung verrichteter Arbeit ergibt sich für die Leistung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">:

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} W=-\sum _{i=1}^{n}Q_{i}^{*}\mathrm {d} q_{i}=-\sum _{i,j=1}^{n}k_{ij}{\dot {q}}_{i}{\dot {q}}_{j}\mathrm {d} t=-2{\mathcal {F}}\mathrm {d} t\Rightarrow {\frac {\mathrm {d} W}{\mathrm {d} t}}=P=-2{\mathcal {F}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8125e3c1fbfefb5dbf070f5462a074703ba8ed3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:68.698ex; height:7.176ex;" alt="{\displaystyle \mathrm {d} W=-\sum _{i=1}^{n}Q_{i}^{*}\mathrm {d} q_{i}=-\sum _{i,j=1}^{n}k_{ij}{\dot {q}}_{i}{\dot {q}}_{j}\mathrm {d} t=-2{\mathcal {F}}\mathrm {d} t\Rightarrow {\frac {\mathrm {d} W}{\mathrm {d} t}}=P=-2{\mathcal {F}}}" loading="lazy"></dd></dl>

wobei die generalisierten Kräfte beschrieben sind durch

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{i}^{*}=-{\frac {\partial {\mathcal {F}}}{\partial {\dot {q}}_{i}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c05756a25f2f3c3093550dda77f328d8d1cc4592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.13ex; height:6.009ex;" alt="{\displaystyle Q_{i}^{*}=-{\frac {\partial {\mathcal {F}}}{\partial {\dot {q}}_{i}}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Ettore Minguzzi: Rayleigh’s Dissipation Function at Work. In: European Journal of Physics. Band 36, Nr. 3, 2015, 035014, <a href="https://arxiv.org/abs/1409.4041" class="extiw external" title="arxiv:1409.4041">arxiv:1409.4041</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Rayleigh, Some general theorems related to vibration, Proc. London Math. Soc. s1-4, 1877, S. 357–368

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Lord Rayleigh, The theory of Sound, Macmillan 1877, Band 1, Kapitel 4, Paragraph 81

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> So in Goldstein, Klassische Mechanik, 8. Auflage, Aula-Verlag 1985, S. 24

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> <a href="Alessandra_Celletti_(Mathematikerin)" title="Alessandra Celletti (Mathematikerin)">Alessandra Celetti</a>: <cite style="font-style:italic">Conservative and Dissipative Models of Celestial Mechanics</cite>. Springer, Cham 2025, ISBN 978-3-03201865-6, S. 15–18.<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:Rayleighsche+Dissipationsfunktion&rft.au=Alessandra+Celetti&rft.btitle=Conservative+and+Dissipative+Models+of+Celestial+Mechanics&rft.date=2025&rft.genre=book&rft.isbn=9783032018656&rft.pages=15-18&rft.place=Cham&rft.pub=Springer" style="display:none">

</li>

</ol></div><div><div class="zim-footer">

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