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Cartesian to spherical change of variables in 3d phase space

Posted by peeterjoot on February 11, 2013

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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

\begin{aligned}\mathcal{L} &= \frac{1}{{2}} m \left(  \dot{r} \hat{\mathbf{r}} + r \sin\theta \dot{\phi} \hat{\boldsymbol{\phi}} + r \dot{\theta} \hat{\boldsymbol{\theta}} \right)^2 \\ &= \frac{1}{{2}} m \left(  \dot{r}^2 + r^2 \sin^2\theta \dot{\phi}^2 + r^2 \dot{\theta}^2  \right)^2\end{aligned} \hspace{\stretch{1}}(1.0.1)

We read off our canonical momentum

\begin{aligned}p_r &= \frac{\partial {\mathcal{L}}}{\partial {r}} \\ &= m \dot{r}\end{aligned} \hspace{\stretch{1}}(1.0.2a)

\begin{aligned}p_\theta &= \frac{\partial {\mathcal{L}}}{\partial {\theta}} \\ &= m r^2 \dot{\theta}\end{aligned} \hspace{\stretch{1}}(1.0.2b)

\begin{aligned}p_\phi &= \frac{\partial {\mathcal{L}}}{\partial {\phi}} \\ &= m r^2 \sin^2\theta \dot{\phi},\end{aligned} \hspace{\stretch{1}}(1.0.2c)

and can now express the Hamiltonian in spherical coordinates

\begin{aligned}H &= \frac{1}{{2}} m \left(\left( \frac{p_r}{m} \right)^2+ r^2 \sin^2\theta \left( \frac{p_\phi}{m r^2 \sin^2\theta} \right)+ r^2 \left( \frac{p_\theta}{m r^2} \right)\right) \\ &= \frac{p_r^2}{2m} + \frac{p_\phi^2}{2 m r^2 \sin^2\theta} + \frac{p_\theta^2}{2 m r^2}\end{aligned} \hspace{\stretch{1}}(1.0.3)

Now we want to do a change of variables. The coordinates transform as

\begin{aligned}x = r \sin\theta \cos\phi\end{aligned} \hspace{\stretch{1}}(1.0.4a)

\begin{aligned}y = r \sin\theta \sin\phi\end{aligned} \hspace{\stretch{1}}(1.0.4b)

\begin{aligned}z = r \cos\theta,\end{aligned} \hspace{\stretch{1}}(1.0.4c)

or

\begin{aligned}r = \sqrt{x^2 + y^2 + z^2}\end{aligned} \hspace{\stretch{1}}(1.0.5a)

\begin{aligned}\theta = \arccos(z/r)\end{aligned} \hspace{\stretch{1}}(1.0.5b)

\begin{aligned}\phi = \arctan(y/x).\end{aligned} \hspace{\stretch{1}}(1.0.5c)

It’s not too hard to calculate the change of variables for the momenta (verified in sphericalPhaseSpaceChangeOfVars.nb). We have

\begin{aligned}p_r = \frac{x p_x + y p_y + z p_z}{\sqrt{x^2 + y^2 + z^2}}\end{aligned} \hspace{\stretch{1}}(1.0.6a)

\begin{aligned}p_\theta = \frac{(p_x x + p_y y) z - p_z (x^2 + y^2)}{\sqrt{x^2 + y^2}}\end{aligned} \hspace{\stretch{1}}(1.0.6b)

\begin{aligned}p_\phi = x p_y - y p_x\end{aligned} \hspace{\stretch{1}}(1.0.6c)

Now let’s compute the volume element in spherical coordinates. This is

\begin{aligned}d\omega &= dr d\theta d\phi p_r p_\theta p_\phi \\ &= \frac{\partial(r, \theta, \phi, p_r, p_\theta, p_\phi)}{\partial(x, y, z, p_x, p_y, p_z)}dx dy dz dp_x dp_y dp_z \\ &= \begin{vmatrix} \frac{x}{\sqrt{x^2+y^2+z^2}} & \frac{y}{\sqrt{x^2+y^2+z^2}} & \frac{z}{\sqrt{x^2+y^2+z^2}} & 0 & 0 & 0 \\  \frac{x z}{\sqrt{x^2+y^2} \left(x^2+y^2+z^2\right)} & \frac{y z}{\sqrt{x^2+y^2} \left(x^2+y^2+z^2\right)} & -\frac{\sqrt{x^2+y^2}}{x^2+y^2+z^2} & 0 & 0 & 0 \\  -\frac{y}{x^2+y^2} & \frac{x}{x^2+y^2} & 0 & 0 & 0 & 0 \\  \frac{\left(y^2+z^2\right) p_x-x y p_y-x z p_z}{\left(x^2+y^2+z^2\right)^{3/2}} & \frac{\left(x^2+z^2\right) p_y-y \left(x p_x+z p_z\right)}{\left(x^2+y^2+z^2\right)^{3/2}} & \frac{\left(x^2+y^2\right) p_z-z \left(x p_x+y p_y\right)}{\left(x^2+y^2+z^2\right)^{3/2}} & \frac{x}{\sqrt{x^2+y^2+z^2}} & \frac{y}{\sqrt{x^2+y^2+z^2}} & \frac{z}{\sqrt{x^2+y^2+z^2}} \\  \frac{y z \left(y p_x-x p_y\right)-x \left(x^2+y^2\right) p_z}{\left(x^2+y^2\right)^{3/2}} & \frac{x z \left(x p_y-y p_x\right)-y \left(x^2+y^2\right) p_z}{\left(x^2+y^2\right)^{3/2}} & \frac{x p_x+y p_y}{\sqrt{x^2+y^2}} & \frac{x z}{\sqrt{x^2+y^2}} & \frac{y z}{\sqrt{x^2+y^2}} & -\sqrt{x^2+y^2} \\  p_y & -p_x & 0 & -y & x & 0 \\ \end{vmatrix}dx dy dz dp_x dp_y dp_z \\ &= dx dy dz dp_x dp_y dp_z\end{aligned} \hspace{\stretch{1}}(1.0.7)

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.

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