# Nonholonomic Noetherian Symmetries and Integrals of the Routh Sphere and the Chaplygin Ball

*2019, Volume 24, Number 5, pp. 511-524*

Author(s):

**Bustamante M. D., Lynch P.**

The dynamics of a spherical body with a non-uniform mass distribution rolling
on a plane were discussed by Sergey Chaplygin, whose 150th birthday 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 a 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 Noetherian symmetries and corresponding 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.

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 Noetherian symmetries and corresponding 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.

Access to the full text on the Springer website |