Analogues of glacial valley profiles in particle mechanics and in cosmology

TL;DR

This paper explores mathematical analogies between glacial valley profiles, particle mechanics, and cosmology, revealing new insights into singularities and universe models through differential equations.

Contribution

It introduces a variational principle-based differential equation linking glacial valleys, particle mechanics, and cosmology, uncovering novel singularity behaviors.

Findings

Analogy with point particle mechanics completes valley profile solutions.

Analogy with Friedmann cosmology reveals a universe with a future singularity.

Identifies a Big Freeze singularity in positive curvature models.

Abstract

An ordinary differential equation describing the transverse profiles of U-shaped glacial valleys, derived with a variational principle, has two formal analogies which we analyze. First, an analogy with point particle mechanics completes the description of the solutions. Second, an analogy with the Friedmann equation of relativistic cosmology shows that the analogue of a glacial valley profile is a universe with a future singularity but respecting the weak energy condition. The equation unveils also a Big Freeze singularity, which was not observed before for positive curvature index.