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:
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.
Read more about this topic: Dynamical Billiards
Famous quotes containing the word motion:
“It may be possible to do without dancing entirely. Instances have been known of young people passing many, many months successively, without being at any ball of any description, and no material injury accrue either to body or mind; Mbut when a beginning is made—when felicities of rapid motion have once been, though slightly, felt—it must be a very heavy set that does not ask for more.”
—Jane Austen (1775–1817)