A geometric interpretation of integrable motions

TL;DR

This paper explores integrable motions in celestial mechanics through a Riemannian geometric lens, linking constants of motion to Killing tensor fields and revealing hidden symmetries in specific models.

Contribution

It introduces a geometric framework connecting integrability to Killing tensor fields, providing explicit expressions for models like Toda and Henon-Heiles.

Findings

Killing tensor fields correspond to hidden symmetries in mechanical manifolds.

Explicit expressions for conserved quantities are derived for specific models.

The geometric approach offers new insights into integrability in celestial mechanics.

Abstract

Integrability, one of the classic issues in galactic dynamics and in general in celestial mechanics, is here revisited in a Riemannian geometric framework, where newtonian motions are seen as geodesics of suitable ``mechanical'' manifolds. The existence of constants of motion that entail integrability is associated with the existence of Killing tensor fields on the mechanical manifolds. Such tensor fields correspond to hidden symmetries of non-Noetherian kind. Explicit expressions for Killing tensor fields are given for the N=2 Toda model, and for a modified Henon-Heiles model, recovering the already known analytic expressions of the second conserved quantity besides energy for each model respectively.