Papers › Assigning probabilities to non-Lipschitz mechanical systems
Assigning probabilities to non-Lipschitz mechanical systems
Danny E. P. Vanpoucke, Sylvia Wenmackers
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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: (d²r(t))/dt² = rᵃ, r(t=0) =0, and (dr(t))/dt|ᵣ₍ₜ₌₀₎ =0, with a ∈]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.
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