Angular Momenta of Relative Equilibrium Motions and Real Moment Map Geometry

TL;DR

This paper explains how the convexity of angular momenta spectra in rigid motions derives from symplectic geometry and provides a representation theoretic description of the real moment map for symmetric pairs.

Contribution

It connects a geometric convexity result with symplectic geometry and offers a new representation theoretic perspective on the real moment map.

Findings

Convex polytope of spectra of angular momenta established.

Connection between rigid motions and symplectic convexity theorem clarified.

Representation theoretic description of the moment map provided.

Abstract

Chenciner and Jimenez Perez showed that the range of the spectra of the angular momenta of all the rigid motions of a fixed central configuration in a general Euclidean space form a convex polytope. In this note we explain how this result follows from a general real convexity theorem of O Shea and Sjamaar in symplectic geometry. Finally, we provide a representation theoretic description of the pushforward of the normalized measure under the real moment map for Riemannian symmetric pairs.