Generalization of the Goryachev-Chaplygin Case

TL;DR

This paper extends classical integrable cases in rigid body dynamics to new settings, including Poisson brackets and quaternion formulations, offering insights into quantum mechanics and force field applications.

Contribution

It introduces a generalized Goryachev-Chaplygin case on Poisson brackets and quaternion equations, expanding the scope of integrable rigid body problems.

Findings

New integrable cases for Poisson brackets and quaternion equations

Invariant relations in generalized rigid body problems

Potential applications in quantum mechanics and force fields

Abstract

In this paper we present a generalization of the Goryachev-Chaplygin integrable case on a bundle of Poisson brackets, and on Sokolov terms in his new integrable case of Kirchhoff equations. We also present a new analogous integrable case for the quaternion form of rigid body dynamics' equations. This form of equations is recently developed and we can use it for the description of rigid body motions in specific force fields, and for the study of different problems of quantum mechanics. In addition we present new invariant relations in the considered problems.