## Velocity volume element to momentum volume element

Posted by peeterjoot on April 9, 2013

# Motivation

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.

# Guts

We are working with a Hamiltonian

where the rest energy is

Hamilton’s equations give us

or

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

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