# Integration: Action Variable J

1. Sep 22, 2009

### quantumkiko

1. The problem statement, all variables and given/known data
A particle of mass m moves in one dimension subject to the potential

$$V = \frac{a}{\sin^2(x/x_0)}$$

Obtain an integral expression for Hamilton's characteristic function. Under what conditions can action-angle variables be used? Assuming these are met, find the frequency of oscillation by the action-angle method. (The integral for J can be evaluated by manipulating the integrand so that the square root appears in the denominator.)

2. The relevant equations

I set up a Hamiltonian

$$H = \frac{p^2}{2m} + \frac{a}{\sin^2(x/x_0)}$$.

Since it doesn't depend on time explicitly, then I can express Hamilton's principal function S as

$$S = W(x, \alpha) + \alpha t$$.

The Hamilton-Jacobi equation is then given by

$$\frac{1}{2m} \left( \frac{\partial W}{\partial x} \right)^2 + \frac{a}{\sin^2(x/x_0)} = \alpha$$

3. The attempt at a solution

a.) Obtain an integral solution for Hamilton's characteristic function.

Isolating the partial derivative of the characteristic function W, I get an integral expression

$$W = \int \sqrt{2m\alpha - \frac{2ma}{\sin^2(x/x_0)}} dx$$.

b.) Under what conditions can action-angle variables be used?

The condition that must be met is $$\alpha \geq \frac{a}{\sin^2(x/x_0)}}$$.

c.) Assuming these are met, find the frequency of oscillation by the action-angle method. (The integral for J can be evaluated by manipulating the integrand so that the square root appears in the denominator.)

For finding the frequency of oscillation, I'm having a problem with integrating the action variable J:

$$J = \int p dx = \int \sqrt{2m\alpha - \frac{2ma}{\sin^2(x/x_0)}} dx$$.

I tried to follow the hint, so I got

$$J = \int \frac{2m\alpha - \frac{2ma}{\sin^2(x/x_0)}}{\sqrt{2m\alpha - \frac{2ma}{\sin^2(x/x_0)}}} dx$$.

Still, I'm stuck. How do I integrate this? Thanks!