Velocity volume element to momentum volume element 1.1

Motivation

One of the problems I attempted had integrals over velocity space with volume element d3 u. 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. 1.2

Guts

We are working with a Hamiltonian e=

q

(pc)2 + e02 ,

(1.1)

where the rest energy is e0 = mc2 .

(1.2)

Hamilton’s equations give us uα = or pα = p

p α / c2 , e muα 1 − u2 / c2

(1.3) .

(1.4)

This is enough to calculate the Jacobian for our volume element change of variables ∂(u x , uy , uz ) dp x ∧ dpy ∧ dpz ∂(p x , py , pz ) 2 2 m c + p2y + p2z − p p − p p y x z x 1 2 2 2 2 dp x ∧ dpy ∧ dpz − p x py m c + p x + pz − pz py = 9/2 6 2 2 2 2 2 2 c m + (p/c) − p x pz − py pz m c + p x + py −5/2 = m2 m2 + p2 / c2 dp x ∧ dpy ∧ dpz . (1.5)

du x ∧ duy ∧ duz =

1

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 d3 u =

c6 e02 3 d p. e5

2

(1.6)