A Note on Norton's Dome

TL;DR

This paper examines Norton's Dome, a Newtonian system with non-unique solutions due to Lipschitz condition violation, reformulating it in a distributional framework to clarify the source of indeterminism.

Contribution

It introduces a weak (distributional) reformulation of Norton's Dome to better understand the indeterminism caused by initial condition interpretation issues.

Findings

Distributional reformulation clarifies indeterminism

Highlights limitations of pointwise initial conditions

Provides a new perspective on non-uniqueness in Newtonian systems

Abstract

"Norton's Dome" is an example of a Newtonian system that violates the Lipschitz condition at a single point, leading to non-unique solutions (indeterminism). Here we reformulate this problem into a "weak" form (in the sense of distributions). In our description the indeterminism manifests through the problematic interpretation of initial conditions, since distributions (as linear functionals on the space of test functions) do not have values at individual points.