Cartesian to spherical change of variables in 3d phase space
Posted by peeterjoot on February 11, 2013
Question: Cartesian to spherical change of variables in 3d phase space
 problem 2.2 (a). Try a spherical change of vars to verify explicitly that phase space volume is preserved.
Our kinetic Lagrangian in spherical coordinates is
We read off our canonical momentum
and can now express the Hamiltonian in spherical coordinates
Now we want to do a change of variables. The coordinates transform as
It’s not too hard to calculate the change of variables for the momenta (verified in sphericalPhaseSpaceChangeOfVars.nb). We have
Now let’s compute the volume element in spherical coordinates. This is
This also has a unit determinant, as we found in the similar cylindrical change of phase space variables.
 RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.