Canonical transformation and generating function

Question

I am stuck on an exercise of which I do not understand the solution. The exercise is the following:

Consider the Hamiltonian system $$H(\theta, p, t) = \dfrac{(p-\omega t)^2}{2} - k\cos(\theta)$$ Find a generating function for a canonical transformation that completes the relation $$P = p-\omega t$$ and calculate the new Hamiltonian and find $k$ such that an elliptic equilibrium point does exit.

Solution

A generating function of second type can be written as

$$F(\theta, P) = \theta(P + \omega t) ~~~~~~~~~~~~~~~~~ \textbf{why??}$$

and the hew Hamiltonian becomes

$$H(\phi, P) = \dfrac{P^2}{2} - k\cos\phi + \omega \phi ~~~~~~~~~~~~~ \textbf{why??}$$

The fixed point can be found by solving the equations

$$P = 0 ~~~~~~~~~~~~~ -k\sin\phi = \omega ~~~~~~~~~~~~~~~~~ \textbf{why??}$$

hence the condition for an elliptic point is $-1 < \omega / k < 1$.

I really need some calrifications to understand those points...

Thank you!

Answer 1

Clearly, there is a Type-II generating function, $F = F_2(\theta, P)$, involved. The transformed "position" coordinate $\phi$ and the old "momentum" coordinate $p$ are related to partial differentials of $F_2$: $$p = \frac{\partial F_2 (\theta, P)}{\partial \theta}; \, \phi = \frac{\partial F_2 (\theta, P)}{\partial P}.$$ You can solve the first of these generally using $p$ in terms of $P$ to get a general expression for $F_2$. After that, use it in the second equation to get $\phi$. Replacing the old coordinates by the new ones should be straightforward: that should give you the new Hamiltonian.

Fixed points are simply points in phase-space where the position and/or the momentum are (is) zero.