On the Cases of Kirchhoff and Chaplygin of the Kirchhoff Equations

TL;DR

This paper investigates the integrability of Kirchhoff and Chaplygin cases in Kirchhoff equations, demonstrating non-integrability in the general Kirchhoff case and analyzing similar behaviors in Chaplygin case, including extensions to higher dimensions.

Contribution

It proves the non-algebraic integrability of the general Kirchhoff case for B ≠ 0 and explores analogous behaviors in the Chaplygin case, extending to four-dimensional analogues.

Findings

The general Kirchhoff case with B ≠ 0 is not algebraically completely integrable.

The Chaplygin case exhibits similar analytic behavior in solutions.

Four-dimensional analogues are constructed on e(4) with Lie-Poisson brackets.

Abstract

It is proven that the general Kirchhoff case of the Kirchhoff equations for $B\ne 0$ is not algebraic complete integrable system. Similar analytic behavior of the general solution of the Chaplygin case is detected. Four-dimensional analogues of the Kirchhoff and the Chaplygin cases are defined on $e(4)$ with the standard Lie-Poisson bracket.