On gliding Lagrange top equations and their asymptotic behaviour

TL;DR

This paper analyzes the asymptotic behavior of the non-integrable gliding Lagrange top, showing that all solutions tend to one of two stable vertical spinning states, with conditions for their stability explicitly determined.

Contribution

It provides a detailed stability analysis of the gliding Lagrange top and characterizes the asymptotic behavior of its solutions, which was previously not fully understood.

Findings

Solutions asymptotically approach stable vertical spinning states

Conditions for stability of these spinning solutions are explicitly derived

The system's non-integrability is highlighted and analyzed

Abstract

The dynamical equations for a gliding Lagrange top are not integrable. They have 5 dynamical variables and admit one integral of motion. We show that all solutions go to one of the two vertical spinning solutions and determine conditions of their stability. This means that solutions starting close to either of the spinning solutions go asymptotically to this solution.