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Hamiltonsches Prinzip
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auch Prinzip der kleinsten Wirkung genannt * Variation der ganzen Bahn im Konfigurationsraum <> Gegensatz d'Ambertsches Prinzip * Wirkung (S) wird extrenmal (minimal) <math>\delta S =0</math> * Start und Zielpunkt <math>(q,t)</math> sind fest vorgegeben (hier keine Variation) * Zeit wird nicht mitvarieiert <math>\delta t =0</math> * Vergleich ART Teilchen Bewegt sich auf Geodäten <> aber nicht im [[Ereignisraum]] * <math>\underline{q}\left( t \right),\underline{q'}\left( t \right)\in {{C}^{2}}</math> (2 fach stetig diffb. Funktionen) * unabhängig von Koordinatenwahl * 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> == spezielle Form== * holonome [[Zwangsbedingungen]] → generalisierte Koordinaten * 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> [[FragenID::M1]] =Herleitung der Euler-Lagrange-Gleichungen= :<math>\begin{align} \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} \end{align}</math> 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} \\ & =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} \\ & =-\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( {{\partial }_{q}}L\delta q+{{\partial }_{{\dot{q}}}}L\delta \dot{q} \right)dt} \end{align}</math> mit partieller Integration (<math>\int{u'v=uv-\int{v'u}}</math>) mit :<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>\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} \\ & =\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( {{d}_{t}}{{\partial }_{{\dot{q}}}}-{{\partial }_{q}} \right)L\delta qdt} \end{align}</math> :<math>\left( {{d}_{t}}{{\partial }_{{\dot{q}}}}-{{\partial }_{q}} \right)L=0</math> [[FrageID::M2]] [[Kategorie:Mechanik]]
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