Explicit integration of one problem of motion of the generalized Kowalevski top

TL;DR

This paper explicitly solves the equations of motion for a specific invariant submanifold of the generalized Kowalevski top in a double force field, using elliptic functions and algebraic expressions.

Contribution

It introduces a method to separate variables and explicitly integrate the motion equations on a particular invariant manifold of the Kowalevski top.

Findings

Equations of motion are separable via a change of variables involving elliptic functions.

Natural phase variables are expressed explicitly in algebraic functions of the new variables.

The approach provides an explicit integration method for this problem.

Abstract

In the problem of motion of the Kowalevski top in a double force field the 4-dimensional invariant submanifold of the phase space was pointed out by M.P.Kharlamov (Mekh. Tverd. Tela, 32, 2002). We show that the equations of motion on this manifold can be separated by the appropriate change of variables, the new variables s1, s2 being elliptic functions of time. The natural phase variables (components of the angular velocity and the direction vectors of the forces with respect to the movable basis) are expressed via s1, s2 explicitly in elementary algebraic functions.