The separation of variables and bifurcations of first integrals in one problem of D.N.Goryachev

TL;DR

This paper analyzes a specific integrable case in rigid body dynamics, using variable separation to derive Abel-Jacobi equations and explicitly express phase variables, thereby describing the system's phase topology.

Contribution

It introduces a real variable separation method for a Goryachev case, deriving explicit algebraic expressions and analyzing phase topology.

Findings

Derived Abel-Jacobi equations with degree 6 polynomial under radical

Expressed phase variables explicitly in elementary algebraic functions

Described the phase topology of the integrable system

Abstract

The equations of motion in one partial integrable case of D.N.Goryachev in the rigid body dynamics are separated by the real change of variables. We obtain the Abel--Jacobi equations with the polynomial of degree 6 under the radical. The natural phase variables (components of the momentum and the momentum force) are expressed via separated variables explicitly in elementary algebraic functions. Basing on these expressions the phase topology of the system is described.