@article{VanpouckeDEPWenmackersS:2021bChaos,
author = {Danny E.P. Vanpoucke and Sylvia Wenmackers},
title = {Assigning probabilities to non-Lipschitz mechanical systems},
journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
volume = {31},
number = {12},
pages = {123131},
year = {2021},
issn = {1089--7682},
doi = {10.1063/5.0063388},
url = {http://dx.doi.org/10.1063/5.0063388},
keywords = {Lipschitz continuity, Newtonian mechanics, Probability theory, Numerical differentiation, Stochastic processes, Classical mechanics, Recurrence relations, Finite difference methods},
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: $\ddot{r} = r^{a}$, $ r(t = 0) = 0$, and $ \dot{r}|_{r(t=0)}= 0$, with $š¯‘ˇ\in ]0,1[$.
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 š¯‘ˇ=1/2 is known as Nortonā€™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.}
}