Nonholonomic Noetherian symmetries and integrals of the Routh sphere and Chaplygin ball

TL;DR

This paper demonstrates how Noether's theorem explains the integrals of motion for the Routh sphere and Chaplygin ball, revealing the geometric and physical significance of these conserved quantities.

Contribution

It derives the Noether symmetries and integrals for the Routh sphere and Chaplygin ball, clarifying their physical meaning and providing a unified symmetry-based approach.

Findings

Derived explicit symmetry transformations for the Routh sphere.

Connected known integrals to Noether's theorem, enhancing understanding.

Extended the method to the Chaplygin ball on a rotating turntable.

Abstract

The dynamics of a spherical body with a non-uniform mass distribution rolling on a plane were discussed by Sergey Chaplygin, whose 150th anniversary we celebrate this year. The Chaplygin top is a non-integrable system, with a colourful range of interesting motions. A special case of this system was studied by Edward Routh, who showed that it is integrable. The Routh sphere has centre of mass offset from the geometric centre, but it has an axis of symmetry through both these points, and equal moments of inertia about all axes orthogonal to the symmetry axis. There are three constants of motion: the total energy and two quantities involving the angular momenta. It is straightforward to demonstrate that these quantities, known as the Jellett and Routh constants, are integrals of the motion. However, their physical significance has not been fully understood. In this paper, we show how the integrals of the Routh sphere arise from Emmy Noether's invariance identity. We derive expressions for the infinitesimal symmetry transformations associated with these constants. We find the finite version of these symmetries and provide their geometrical interpretation. As a further demonstration of the power and utility of this method, we find the Noether symmetries and corresponding Noether integrals for a system introduced recently: the Chaplygin ball on a rotating turntable, confirming that the known integrals are directly obtained from Noether's theorem.