[QUE/CM-10007]

1. What are the conditions on ``small'' constants \(a,b,c,d,e\) and \(f\) in order that \begin{eqnarray} q= Q + a Q^2 +2b QP + cP^2\\ p= P + d Q^2 + 2e QP + fP^2 \end{eqnarray} be a canonical transformation to the first order in small quantities?
2. The Hamiltonian for a slightly anharmonic oscillator is \[H=\frac{p^2}{2m} + \frac{1}{2}m\omega^2q^2 \beta q^3\] where \(\beta\) is ``small.'' Perform a canonical transformation of the type given given in part (a) and adjust the constants so that the new Hamiltonian \(H\) does not contain an anharmonic term to first order in small quantities, thus \[\bar{H} = \frac{P^2}{2m} + \frac{1}{2}m \omega^2 Q^2 + \text{second order terms.}\]
3. Write down and solve Hamilton's equations for the new canonical variables, and then use the transformation equations to find the solution to the anharmonic oscillator problem valid to first order in small quantities.

Source: Calkin