The problem of the motion of a solid in an ideal fluid. Integration of the Clebsch's case

Abstract

This paper deals with a geometric approach to the integration of Clebsch's case of equations describing the motion of a solid body in an ideal fluid. This problem is defined by a nonlinear system of 6 differential equations admitting 4 polynomial first integrals. We show that the intersection of surface levels of these integrals can be completed to an abelian surface, i.e., a 2-dimensional algebraic torus. Also, we prove that the problem can be linearized, i.e., can be written in terms of abelian integrals, on a Prym variety \( \ ym_{\tau } (\mathcal C) \) of a genus 3 curve \( \mathcal{C} \) obtained naturally.