Simple pendulum phase space

1. Feb 4, 2009

shehry1

1. The problem statement, all variables and given/known data
Pathria 2.6 (2nd Edition): Phase space volume of a simple pendulum.

The total energy can be expressed in the form of the time derivative of the angle + the Sin^2 of that angle.

From this I want to calculate the phase space volume. Mathematica gives the solution in the form of elliptic integrals. So I used the Simple pendulum angle limit ie: Six(x) = x. But because of this I cannot introduce a pi into the equation as by doing so, the limits of the phase space volume should change to x_max and -x_max. Is that correct?

Or can I still use the 0 to 2(Pi) limits of integration. I did a quick calculation and the answer comes correct when I use both the small angle approximation and integrate over the (0, 2(pi)) limit. But it seems wrong somehow.

2. Relevant equations
Total energy of a simple pendulum

3. The attempt at a solution

Thanks