Two dimensional curved disks on a sphere: the evolution of kinetic energy

TL;DR

This paper investigates how disks on a spherical surface interact and how their total kinetic energy evolves, revealing an increase due to non-radial velocity components, which differs from flat space behavior.

Contribution

It introduces a model of disks on a sphere with curved geometry and analyzes their kinetic energy evolution, highlighting effects unique to curved spaces.

Findings

Kinetic energy increases over time on a sphere.

Non-radial velocity components drive energy growth.

Behavior differs from flat space granular systems.

Abstract

We put disks on a sphere between two parallels of this sphere. The disks have the curvature of the sphere and interact via a simplified Hertz law. We analyze the behavior of the total kinetic energy of the whole assembly of disks with Riemanian geometry and contrarily to flat spaces, this kinetic energy increases. This is due to a non radial component of the resulting velocity vector at each interaction between two disks. The study of granular matter in twodimensional curved spaces is therefore not trivial.