In teaching a course in classical mechanics and I have come across (from my predecessor) a to me mysterious conserved quantity.

We are considering a gravitational (or electric) potential with the Hamiltonian $$ H = \frac{p_r^2}{2m} + \frac{p_\theta^2}{2mr^2} - \frac{k}{r}. $$

The quantity in question is $$ A_x = \frac{p_\theta^2\cos\theta}{r} - mk\cos\theta + p_rp_\theta\sin\theta$$

Using Poisson brackets you can show that $\{H,A_x\}=0$ so $A_x$ is conserved.

What puzzles me is that it has dimensions of J*m*kg so it's is not a strange linear combination of angular momenta. It also explicitly contains the potential strength $k$, which looks a bit odd.

My question is simply, what is the interpretation of $A_x$?


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