Lagrange top: integrability according to Liouville and examples of analytic solutions

TL;DR

This paper revisits the equations of motion for the Lagrange top with a more accurate potential energy formulation, demonstrating its integrability and providing new analytical solutions, especially in the case of precession without nutation.

Contribution

It introduces an improved formulation of the Lagrange top equations with a diagonal inertia tensor and presents new analytical solutions, enhancing understanding of its integrability.

Findings

The improved equations are integrable according to Liouville.

Several new analytical solutions are provided.

Precession without nutation exhibits a complex relationship between rotation and precession rates.

Abstract

Equations of a heavy rotating body with one fixed point can be deduced starting from a variational problem with holonomic constraints. When applying this formalism to the particular case of a Lagrange top, in the formulation with a diagonal inertia tensor the potential energy has more complicated form as compared with that assumed in the literature on dynamics of a rigid body. This implies the corresponding improvements in equations of motion. Therefore, we revised this case, presenting several examples of analytical solutions to the improved equations. The case of precession without nutation has a surprisingly rich relationship between the rotation and precession rates, and this is discussed in detail.