Velocity volume element to momentum volume element
Posted by peeterjoot on April 9, 2013
One of the problems I attempted had integrals over velocity space with volume element . Initially I thought that I’d need a change of variables to momentum space, and calculated the corresponding momentum space volume element. Here’s that calculation.
We are working with a Hamiltonian
where the rest energy is
Hamilton’s equations give us
This is enough to calculate the Jacobian for our volume element change of variables
That final simplification of the determinant was a little hairy, but yielded nicely to Mathematica.
Our final result for the velocity volume element in momentum space, in terms of the particle energy is