Editing Kanonische Transformationen
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Dies ist also die identische Transformation | Dies ist also die identische Transformation | ||
====Beispiel: Harmonischer Oszillator:==== | |||
:<math>\begin{align} | |||
& H=\frac{{{p}^{2}}}{2m}+\frac{m{{\omega }^{2}}}{2}{{q}^{2}} \\ | |||
& {{M}_{1}}(q,Q)=\frac{m\omega }{2}{{q}^{2}}\cot Q \\ | |||
& \Rightarrow p=\frac{\partial {{M}_{1}}}{\partial q}=m\omega q\cot Q \\ | |||
& P=-\frac{\partial {{M}_{1}}}{\partial Q}=\frac{m\omega }{2}\frac{{{q}^{2}}}{{{\sin }^{2}}Q} \\ | |||
& q={{\left( \frac{2}{m\omega }P \right)}^{\frac{1}{2}}}\sin Q \\ | |||
& p={{\left( 2m\omega P \right)}^{\frac{1}{2}}}\cos Q \\ | |||
& \\ | |||
\end{align}</math> | |||
:<math>\begin{align} | |||
& H=\bar{H}\quad \left( \frac{\partial {{M}_{1}}}{\partial t} \right)=0 \\ | |||
& H=\frac{2m\omega P{{\cos }^{2}}Q}{2m}+\frac{m{{\omega }^{2}}2P}{2m\omega }{{\sin }^{2}}Q=\omega P \\ | |||
\end{align}</math> | |||
Die Variable Q ist also zyklisch. | |||
:<math>\begin{align} | |||
& \dot{P}=-\frac{\partial H}{\partial Q}=0\Rightarrow P=\alpha =const \\ | |||
& \dot{Q}=\frac{\partial H}{\partial P}=\omega \Rightarrow Q=\omega t+\beta \\ | |||
\end{align}</math> | |||
Somit kann q(t) durch Integration (2 Integrationskonstanten!!) gefunden werden: | |||
:<math>q(t)={{\left( \frac{2\alpha }{m\omega } \right)}^{\frac{1}{2}}}\sin \left( \omega t+\beta \right)</math> | |||
Dabei beschreibt | |||
:<math>\alpha </math> | |||
die Amplitude und | |||
:<math>\beta </math> | |||
die Phase. |