Assigning probabilities to non-Lipschitz mechanical systems

Abstract

We present a method for assigning probabilities to the solutions of initial value problems that have a Lipschitz singularity. To illustrate the method, we focus on the following toy example: d(2)r(t)/dt(2) = r(a), r ( t = 0 ) = 0, and d r ( t )/d t divide r ( t = 0 ) = 0, with a & ISIN; ] 0 , 1 [. This example has a physical interpretation as a mass in a uniform gravitational field on a frictionless, rigid dome of a particular shape; the case with a = 1 / 2 is known as Norton's dome. Our approach is based on (1) finite difference equations, which are deterministic; (2) elementary techniques from alpha-theory, a simplified framework for non-standard analysis that allows us to study infinitesimal perturbations; and (3) a uniform prior on the canonical phase space. Our deterministic, hyperfinite grid model allows us to assign probabilities to the solutions of the initial value problem in the original, indeterministic model.