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This is for a course focusing on the connections between Newtonian, Lagrangian and Hamiltonian formalisms.

We're given an elliptic billiard table with foci 1 and 2, where $L_1$ and $L_2$ are the angular momentums with respect to the two foci. Using the Hamiltonian formalism, I need to prove that $L_1\cdot L_2$ is a constant of motion.

So I figured I would need to use Poisson brackets and the fact if $\{ I,H \} =0$, I is a constant of motion. Working backwards, I get to the point where:

$$ \frac{\partial (L_1\cdot L_2)}{\partial q}\frac{\partial H}{\partial p}=\frac{\partial (L_1\cdot L_2)}{\partial p}\frac{\partial H}{\partial q} $$

So I thought I would start with the Lagrangian for a free moving particle, work in the constraints for the ellipse, then work out how to write the angular momentum scalar product in terms of $p$ and $q$. Hopefully, doing this and then going to the Hamiltonian will let me prove the above.

What I'm looking for is confirmation (or not) that I'm on the right track, and where I should go from here. I'm also having some trouble with how I would work in the constraint regarding the ellipse boundary.

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I think the key point would be the geometrical fact that, at a point at the border of an ellipse, the bisector of the two lines passing through that point and the focal points, is perpendicular to the border! There is a nice physical proof for this theorem, using the Fermat's principle for light! – Ali May 29 '13 at 17:34

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