Assigning probabilities to non-Lipschitz mechanical systems

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: $\frac{d^2r(t)}{dt^2} = r^a$, $r(t=0) =0$, and $\frac{dr(t)}{dt}\mid_{r(t=0)} =0$, with $a \in ]0,1[$. 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.