Miguel Bustamante

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.

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock’n’roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.