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

TL;DR

By lifting the heavy-top dynamics to the 3-sphere $S^3$ with a left-invariant metric from the inertia tensor, the Euler--Poisson system is recast as a curvature-balance problem between inertial geodesic curvature and gravity forcing. For the inertia choice $I=(2,2,1)$ and center-of-mass direction $r=(0,0,1)$, a distinguished pure--precession regime emerges where $\gamma_3$ and $\omega_3$ stay constant and the horizontal components follow an isochronous linear system with explicit trig solutions, complemented by a Lax representation for the reduced dynamics. Although the full heavy-top system is not globally integrable, this regime provides a geometrically natural, dynamically simplified subsystem, illuminating how curvature lifting can reveal hidden structure in non-integrable mechanics. A curvature-forcing indicator based on $F(t)=\|\gamma\times r\|$ guided the identification of the $(2,2,1)$ case, illustrating how geometric methods can uncover partial symmetries and reductions valuable for analysis and potential generalization.

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.