Dynamical Billiards - Equations of Motion

Equations of Motion

The Hamiltonian for a particle of mass m moving freely without friction on a surface is:

where is a potential designed to be zero inside the region in which the particle can move, and infinity otherwise:

V(q)=begin{cases} 0 qquad q in Omega, \ infty qquad q notin Omega. end{cases}

This form of the potential guarantees a specular reflection on the boundary. The kinetic term guarantees that the particle moves in a straight line, without any change in energy. If the particle is to move on a non-Euclidean manifold, then the Hamiltonian is replaced by:

where is the metric tensor at point . Because of the very simple structure of this Hamiltonian, the equations of motion for the particle, the Hamilton–Jacobi equations, are nothing other than the geodesic equations on the manifold: the particle moves along geodesics.

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