The (2,2,1) heavy top: a pure-precession regime

TL;DR

This paper introduces a geometric curvature-based approach to analyze the Euler-Poisson equations for a specific inertia ratio, revealing a unique pure-precession regime with explicit solutions and a Lax representation, highlighting geometric structures in complex dynamics.

Contribution

It develops a novel curvature-based geometric formulation and explicitly characterizes a pure-precession regime for the (2,2,1) inertia ratio, including a Lax representation and symmetry detection methods.

Findings

Identifies a pure-precession regime with constant tilt angle.

Derives explicit trigonometric solutions for the regime.

Uses curvature forcing to detect geometrically significant parameters.

Abstract

This work develops a curvature-based geometric formulation of the Euler-Poisson equations by lifting the dynamics to the 3-sphere S^3 equipped with the left-invariant metric induced by the inertia tensor. For the inertia ratio I = (2,2,1) and r = (0,0,1), the curvature balance reveals a distinguished pure-precession regime: a nontrivial family of motions in which the tilt angle gamma_3 remains constant and the dynamics reduce to uniform precession with explicit trigonometric solutions. The family is characterized and derived explicitly, and a Lax representation is obtained. This regime illustrates how geometric lifting and curvature balance can isolate simplified dynamical structures even inside non-integrable systems. In addition, we briefly discuss the role of a numerical symmetry detection procedure based on curvature forcing, which guided the identification of the (2,2,1) parameters as geometrically distinguished.