The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions

TL;DR

This paper investigates the integrability of Poisson equations in the nonholonomic Suslov problem, deriving conditions for meromorphic solutions, explicit solutions for special cases, and analyzing the hypergeometric form and monodromy to understand the system's behavior.

Contribution

It provides necessary and sufficient conditions for meromorphic solutions, explicit solutions for special motions, and links the equations to hypergeometric functions, advancing understanding of the Suslov problem's integrability.

Findings

Derived conditions for meromorphic solutions.

Explicit solutions for special motions.

Analyzed hypergeometric form and monodromy group.

Abstract

We consider the problem of integrability of the Poisson equations describing spatial motion of a rigid body in the classical nonholonomic Suslov problem. We obtain necessary conditions for their solutions to be meromorphic and show that under some further restrictions these conditions are also sufficient. The latter lead to a family of explicit meromorphic solutions, which correspond to rather special motions of the body in space. We also give explicit extra polynomial integrals in this case. In the more general case (but under one restriction), the Poisson equations are transformed into a generalized third order hypergeometric equation. A study of its monodromy group allows us also to calculate the "scattering" angle: the angle between the axes of limit permanent rotations of the body in space.