Dynamics on strata of trigonal Jacobians and some integrable problems of rigid body motion

TL;DR

This paper analyzes the Goryachev case of rigid body motion, revealing its complex integrable structure involving dynamics on trigonal Jacobian strata and introducing new sigma-function solutions for non-hyperelliptic curves.

Contribution

It provides the first detailed algebraic geometric and analytical description of the Goryachev case involving trigonal Jacobians, expanding understanding of integrable systems beyond hyperelliptic cases.

Findings

Solution involves dynamics on strata of trigonal Jacobians

Introduces new sigma-function formulae for non-hyperelliptic curves

First example of an integrable system with solutions on trigonal curves

Abstract

We present an algebraic geometrical and analytical description of the Goryachev case of rigid body motion. It belongs to a family of systems sharing the same properties: although completely integrable, they are not algebraically integrable, their solution is not meromorphic in the complex time and involves dynamics on the strata of the Jacobian varieties of trigonal curves. Although the strata of hyperelliptic Jacobians have already appeared in the literature in the context of some dynamical systems, the Goryachev case is the first example of an integrable system whose solution involves a more general curve. Several new features (and formulae) are encountered in the solution given in terms of sigma-functions of such a curve.