Non-existence of an invariant measure for a homogeneous ellipsoid rolling on the plane

TL;DR

This paper proves that a homogeneous ellipsoid with all different semi-axes rolling on a plane does not admit a smooth invariant measure, contrasting with the symmetric case where such a measure exists.

Contribution

It establishes the non-existence of an invariant measure for the general asymmetric case, extending previous results limited to symmetric ellipsoids.

Findings

Invariant measure exists for axially symmetric ellipsoids

No invariant measure exists for fully asymmetric ellipsoids

Results clarify the dynamics of rolling ellipsoids with different axes

Abstract

It is known that the reduced equations for an axially symmetric homogeneous ellipsoid that rolls without slipping on the plane possess a smooth invariant measure. We show that such an invariant measure does not exist in the case when all of the semi-axes of the ellipsoid have different length.