## Cartesian to spherical change of variables in 3d phase space

Posted by peeterjoot on February 11, 2013

[Click here for a PDF of this post with nicer formatting]

## Question: Cartesian to spherical change of variables in 3d phase space

[1] problem 2.2 (a). Try a spherical change of vars to verify explicitly that phase space volume is preserved.

## Answer

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

or

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.

# References

[1] RK Pathria. *Statistical mechanics*. Butterworth Heinemann, Oxford, UK, 1996.

## cosmas zachos said

The very first line of the huge Jacobian determinant eqn 1.0.7 needs differentials dp for the momenta p. It is fixed in the second line. The partials of the Lagrangian in 1.0.2 need to be w.r.t. the velocities, so dotted quantities.

## peeterjoot said

Thanks Cosmas. Fixed (and will be fixed in the next version of phy452.pdf that I post).

## Someone said

Hey there!

About second equation of 1.0.1, there’s no square in the parenthesis, because the square was already evaluated when going from first equation to second equation.

I hope I could help! :).

## peeterjoot said

Thanks. fixed.