On the dynamics of particles in a bounded region: A measure theoretical approach

Abstract

An existence theorem is proven for the solution of the differential equations of motion of a finite number of particles moving in a bounded piecewise regular region and mutually interacting via C¹ forces. it is shown that the elastic reflection laws uniquely determine a Lebesque measure solution of the differential equations of motion (with elastic boundary conditions). The Lebesgue measure is invariant so that an extension of the Liouville theorem to non‐Hamiltonian flows is obtained. A natural representation of the time evolution is given as a flow upward from a base under a ’’ceiling’’ function.