Hamiltonsches Prinzip - Revision history

>SchuBot: / spezielle Form */Pfeile einfügen, replaced: --> → → (2)

2010-09-12T19:50:14Z

spezielle Form: Pfeile einfügen, replaced: --> → → (2)

← Older revision		Revision as of 21:50, 12 September 2010
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	:<math>\delta S=\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( \delta T-\delta A \right)dt}=0</math> mit <math>\delta A=\sum\limits_{i}{{{\underline{X}}_{i}}\delta \underline{{{r}_{i}}}}</math>		:<math>\delta S=\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( \delta T-\delta A \right)dt}=0</math> mit <math>\delta A=\sum\limits_{i}{{{\underline{X}}_{i}}\delta \underline{{{r}_{i}}}}</math>
	== spezielle Form==		== spezielle Form==
	* holonome [[Zwangsbedingungen]] ~~-->~~ generalisierte Koordinaten		* holonome [[Zwangsbedingungen]] → generalisierte Koordinaten
	* konservative Kräfte ~~-->~~ <math>L=T-V</math>		* konservative Kräfte → <math>L=T-V</math>
	führt zur Wirkung <math>S\left[ q \right]:=\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{L\left( q,\dot{q},t \right)dt}</math>		führt zur Wirkung <math>S\left[ q \right]:=\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{L\left( q,\dot{q},t \right)dt}</math>

*>SchuBot: Einrückungen Mathematik

2010-09-12T15:26:53Z

Einrückungen Mathematik

← Older revision		Revision as of 17:26, 12 September 2010
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	* unabhängig von Koordinatenwahl		* unabhängig von Koordinatenwahl
	* Allgemein		* Allgemein
	<math>\delta S=\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( \delta T-\delta A \right)dt}=0</math> mit <math>\delta A=\sum\limits_{i}{{{\underline{X}}_{i}}\delta \underline{{{r}_{i}}}}</math>		:<math>\delta S=\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( \delta T-\delta A \right)dt}=0</math> mit <math>\delta A=\sum\limits_{i}{{{\underline{X}}_{i}}\delta \underline{{{r}_{i}}}}</math>
	== spezielle Form==		== spezielle Form==
	* holonome [[Zwangsbedingungen]] --> generalisierte Koordinaten		* holonome [[Zwangsbedingungen]] --> generalisierte Koordinaten
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	[[FragenID::M1]]		[[FragenID::M1]]
	=Herleitung der Euler-Lagrange-Gleichungen=		=Herleitung der Euler-Lagrange-Gleichungen=
	<math>\begin{align}		:<math>\begin{align}
	\delta S\left[ q \right] & =\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\delta L\left( q,\dot{q},t \right)dt} \\		\delta S\left[ q \right] & =\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\delta L\left( q,\dot{q},t \right)dt} \\
	& =\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( {{\partial }_{q}}L\delta q+{{\partial }_{{\dot{q}}}}L\delta \dot{q} \right)dt}		& =\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( {{\partial }_{q}}L\delta q+{{\partial }_{{\dot{q}}}}L\delta \dot{q} \right)dt}
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	mit partieller Integration (<math>\int{u'v=uv-\int{v'u}}</math>) mit		mit partieller Integration (<math>\int{u'v=uv-\int{v'u}}</math>) mit
	<math>u=\delta q,v={{\partial }_{{\dot{q}}}}L</math>		:<math>u=\delta q,v={{\partial }_{{\dot{q}}}}L</math>




	<math>{{\partial }_{{\dot{q}}}}L\delta \dot{q}={{d}_{t}}\left( {{\partial }_{{\dot{q}}}}L\delta q \right)-{{d}_{t}}\left( {{\partial }_{{\dot{q}}}}L \right)\delta q</math>		:<math>{{\partial }_{{\dot{q}}}}L\delta \dot{q}={{d}_{t}}\left( {{\partial }_{{\dot{q}}}}L\delta q \right)-{{d}_{t}}\left( {{\partial }_{{\dot{q}}}}L \right)\delta q</math>


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	<math>\begin{align}		:<math>\begin{align}
	\delta S\left[ q \right] & =- \cancel {\left[ {{\partial }_{{\dot{q}}}}L\delta q \right]_{{{t}_{1}}}^{{{t}_{2}}}} -\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( {{\partial }_{q}}L\delta q-{{d}_{t}}\left( {{\partial }_{{\dot{q}}}}L \right)\delta q \right)dt} \\		\delta S\left[ q \right] & =- \cancel {\left[ {{\partial }_{{\dot{q}}}}L\delta q \right]_{{{t}_{1}}}^{{{t}_{2}}}} -\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( {{\partial }_{q}}L\delta q-{{d}_{t}}\left( {{\partial }_{{\dot{q}}}}L \right)\delta q \right)dt} \\
	& =\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( {{d}_{t}}{{\partial }_{{\dot{q}}}}-{{\partial }_{q}} \right)L\delta qdt}		& =\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( {{d}_{t}}{{\partial }_{{\dot{q}}}}-{{\partial }_{q}} \right)L\delta qdt}
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	<math>\left( {{d}_{t}}{{\partial }_{{\dot{q}}}}-{{\partial }_{q}} \right)L=0</math>		:<math>\left( {{d}_{t}}{{\partial }_{{\dot{q}}}}-{{\partial }_{q}} \right)L=0</math>
	[[FrageID::M2]]		[[FrageID::M2]]
	[[Kategorie:Mechanik]]		[[Kategorie:Mechanik]]

*>SchuBot: Einrückungen Mathematik

2010-09-12T15:06:17Z

Einrückungen Mathematik

← Older revision		Revision as of 17:06, 12 September 2010
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	* unabhängig von Koordinatenwahl		* unabhängig von Koordinatenwahl
	* Allgemein		* Allgemein
	<math>\delta S=\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( \delta T-\delta A \right)dt}=0</math>		<math>\delta S=\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( \delta T-\delta A \right)dt}=0</math> mit <math>\delta A=\sum\limits_{i}{{{\underline{X}}_{i}}\delta \underline{{{r}_{i}}}}</math>
	mit
	<math>\delta A=\sum\limits_{i}{{{\underline{X}}_{i}}\delta \underline{{{r}_{i}}}}</math>
	== spezielle Form==		== spezielle Form==
	* holonome [[Zwangsbedingungen]] --> generalisierte Koordinaten		* holonome [[Zwangsbedingungen]] --> generalisierte Koordinaten
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	\delta S\left[ q \right] & =\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\delta L\left( q,\dot{q},t \right)dt} \\		\delta S\left[ q \right] & =\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\delta L\left( q,\dot{q},t \right)dt} \\
	& =\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( {{\partial }_{q}}L\delta q+{{\partial }_{{\dot{q}}}}L\delta \dot{q} \right)dt}		& =\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( {{\partial }_{q}}L\delta q+{{\partial }_{{\dot{q}}}}L\delta \dot{q} \right)dt}
	\end{align}</math>		\end{align}</math> oder <math>\begin{align}

	oder

	<math>\begin{align}
	\delta S\left[ q \right] & =S\left[ {{q}_{0}} \right]-\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{L\left( q+\delta q,\dot{q}+\delta \dot{q},t \right)dt} \\		\delta S\left[ q \right] & =S\left[ {{q}_{0}} \right]-\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{L\left( q+\delta q,\dot{q}+\delta \dot{q},t \right)dt} \\
	& =S\left[ {{q}_{0}} \right]-\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( \underbrace{L}_{=S\left[ {{q}_{0}} \right]}+{{\partial }_{q}}L\delta q+{{\partial }_{{\dot{q}}}}L\delta \dot{q} \right)dt} \\		& =S\left[ {{q}_{0}} \right]-\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( \underbrace{L}_{=S\left[ {{q}_{0}} \right]}+{{\partial }_{q}}L\delta q+{{\partial }_{{\dot{q}}}}L\delta \dot{q} \right)dt} \\

Schubotz: /* Herleitung der Euler-Lagrange-Gleichungen */

2009-07-18T23:07:02Z