# Proving angular momentum is conserved

1. Jan 11, 2009

1. The problem statement, all variables and given/known data
A ball moves in a two-dimensional billiard comprising a horizontal frictionless table enclosed by two circular barriers of radii 2m and 1m and centred, respectively, at (x, y)=(0, 0) and (x, y)=(a, 0), where a<0.5m. The ball undergoes specular reflection at each barrier. Prove that the ball's angular momentum is conserved when a=0.

2. Relevant equations
Poisson brackets

3. The attempt at a solution
I know that I have to bring poisson brackets into this, since it basically comes down to showing constants of motion, although I don't know what I need to use poisson brackets on. I think that I need to consider $L^2$ and the hamiltonian of the system, but I'm not entirely sure how to build the boundary conditions (e.g. specular reflection off the boundary walls) into the hamiltonian.