Hamiltonian reductions of free particles under polar actions of compact Lie groups

TL;DR

This paper explores Hamiltonian reduction of free particles with symmetry groups acting in a polar manner, leading to integrable systems related to spin Calogero-Sutherland models in classical and quantum contexts.

Contribution

It introduces a framework for Hamiltonian reduction under polar actions, connecting geometric symmetry properties to integrable many-body systems.

Findings

Reduced systems are explicitly described under polar actions.

Hyperpolar actions on Lie groups yield integrable spin Calogero-Sutherland systems.

The approach applies to both classical and quantum Hamiltonian systems.

Abstract

Classical and quantum Hamiltonian reductions of free geodesic systems of complete Riemannian manifolds are investigated. The reduced systems are described under the assumption that the underlying compact symmetry group acts in a polar manner in the sense that there exist regularly embedded, closed, connected submanifolds meeting all orbits orthogonally in the configuration space. Hyperpolar actions on Lie groups and on symmetric spaces lead to families of integrable systems of spin Calogero-Sutherland type.