||Message]
Suppose that \(q,p\) defined by
\begin{equation}
Q= q\cos\theta - \frac{p}{m\omega} \sin\theta,\qquad P=m\omega q
\sin\theta + p \cos\theta
\end{equation}
are the canonical variables for a simple harmonic oscillator with Hamiltonian
\[ H=\frac{p^2}{2m} +\frac{1}{2}m \omega^2 q^2\]
- Find the Hamiltonian \(K(Q,P,t)\) for the new canonical variables \(Q,P\), assuming that \(\theta\) is some function of time \(\theta(t)\). Show that we can choose \(\theta(t)\) so that \(K=0\).
- With this choice of \(\theta(t)\) solve the new canonical equations to find \(Q,P\) as functions of time, and then use the transformation equations to find the original variables \(q,p\) as functions of time.
Exclude node summary :
n
Exclude node links:
0
4727:Diamond Point
0
