A Hamiltonian has the form

\[H=q_{1} p_{1}-q_{2} p_{2}+a q_{1}^{2}-b q_{2}^{2}\]

where \(a\) and \(b\) are constants.

(a) Using the method of Poisson brackets, show that

\[f_{1} \equiv q_{1} q_{2} \quad \text { and } \quad f_{2} \equiv \frac{1}{q_{1}}\left(p_{2}+b q_{2}\right)\]

are constants of the motion.

(b) Then show that \(\left\{f_{1}, f_{2}\right\}\) is also a constant of the motion.

(c) Is \(H\) itself constant? Check by finding \(q_{1}, q_{2}, p_{1}\), and \(p_{2}\) as explicit functions of time.