Rigid motions: action-angles, relative cohomology and polynomials with roots on the unit circle

TL;DR

This paper revisits the canonical integration of a classical rigid body near uniform rotation, constructing action-angle coordinates through power series and analyzing polynomial roots on the unit circle, avoiding elliptic integrals.

Contribution

It introduces a novel approach to derive action-angle coordinates using power series and relative cohomology, revealing properties of associated polynomials with roots on the unit circle.

Findings

Action-angle coordinates constructed without elliptic integrals.

Identification of polynomials with roots on the unit circle.

New conjectures about polynomial properties and root locations.

Abstract

Revisiting canonical integration of the classical solid near a uniform rotation, canonical action angle coordinates, hyperbolic and elliptic, are constructed in terms of various power series with coefficients which are polynomials in a variable $r^2$ depending on the inertia moments. Normal forms are derived via the analysis of a relative cohomology problem and shown to be obtainable without the use of ellitptic integrals (unlike the derivation of the action-angles). Results and conjectures also emerge about the properties of the above polynomials and the location of their roots. In particular a class of polynomials with all roots on the unit circle arises.