Chaplygin ball over a fixed sphere: explicit integration

TL;DR

This paper analyzes a generalized nonholonomic rolling sphere system, demonstrating its integrability for specific parameters, and provides explicit solutions using advanced coordinate transformations and theta-functions.

Contribution

It introduces a new integrable case of the nonholonomic Chaplygin sphere problem with explicit solutions and novel separation coordinates.

Findings

System is integrable for a specific ratio of sphere radii.

Explicit integration achieved via separation of variables and theta-functions.

New coordinates generalizing ellipsoidal coordinates are introduced.

Abstract

We consider a nonholonomic system describing a rolling of a dynamically non-symmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel--Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconical) coordinates on the Poisson sphere, which can be useful in other integrable problems. Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.