Non-existence of Physical Classical Solutions to Euler's Equations of Rigid Body Dynamics

TL;DR

This paper proves that for non-spherical rigid bodies, classical solutions to Euler's equations cannot exist globally in time due to the non-existence of collision resolution maps that conserve physical quantities.

Contribution

It demonstrates the fundamental non-existence of classical solutions for non-spherical bodies in rigid body Euler dynamics, highlighting the need to consider infinitely many collisions.

Findings

Classical solutions do not exist for non-spherical bodies in Euler's equations.

Collision resolution maps conserving physical quantities cannot be constructed.

Solutions involve infinitely many collisions in finite time.

Abstract

We prove that one cannot construct, for arbitrary initial data, global-in-time physical classical solutions to Euler's equations of continuum rigid body mechanics when the constituent rigid bodies are not perfect spheres. By 'physical' solutions, we mean those that conserve the total linear momentum, angular momentum and kinetic energy of any given initial datum. The reason for absence of classical solutions is due to the non-existence of velocity scattering maps which resolve a collision between two non-spherical rigid bodies in such a way that (i) they do not interpenetrate, and (ii) total linear momentum, angular momentum and kinetic energy of the bodies are conserved through collision. In particular, this implies that when solving Euler's equations, it is necessary to deal with rigid body trajectories which experience infinitely-many collisions in a finite time interval.