Question 7
Given the following Hamiltonian,
$H(q,p) = \frac{p^2}{2f(q)^2} + \frac{F(q)^2}{2}$ (12)
Transform $(q, p)$ to $(Q, P)$ in a way that
$H(Q, P) = \frac{P^2}{2} + \frac{Q^2}{2}$ (13)
The Hamiltonian above suggests $P = \frac{p}{f(q)}$ and that $Q = F(q)$
(a) By explicitly calculating the Poisson bracket, find the relationship between $f(q)$ and $F(q)$ when $(Q, P)$ is a canonical pair.
(b) Determine Hamilton's equations for the canonical pair $(Q, P)$ in Equation (2) and find the general solution for $(Q, P)$
