Simple pendulum phase space

[shehry1]

Homework Statement

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.

Homework Equations

Total energy of a simple pendulum

The Attempt at a Solution

Thanks

 

[BigJDubs]

for the help! Yes, I am trying to calculate the phase space volume of a simple pendulum. The total energy of a simple pendulum is given by: E = (1/2)*m*l^2*x_dot^2 + m*g*l*(1-cos(x))Where x is the angle, x_dot is the angular velocity, m is the mass of the pendulum and l is the length of the pendulum. I want to calculate the phase space volume given by:V = Integral from x_min to x_max (dx/sqrt(2(E-V(x)))) where V(x) is the potential energy of the system. Using the small angle approximation, the potential energy reduces to: V(x) = m*g*l*x^2/2. Substituting this into the expression for the phase space volume gives:V = Integral from 0 to 2(pi) (dx/sqrt(2(E - m*g*l*x^2/2))) Which gives the solution in terms of elliptic integrals. Since I am using the small angle approximation, it is not necessary to use the -x_max to x_max limits of integration. Is that correct?