The degenerate C. Neumann system I: symmetry reduction and convexity

TL;DR

This paper analyzes the symmetry reduction and convexity properties of the degenerate C. Neumann system, revealing its geometric structure, integrability, and separation of variables, with implications for superintegrability and convexity in Hamiltonian systems.

Contribution

It provides a detailed geometric and algebraic analysis of the degenerate C. Neumann system, including symmetry reduction, embedding of reduced spaces, separation of variables, and convexity results.

Findings

Reduced Neumann system separates in elliptical-spherical coordinates.

Action variables are expressed as hyperelliptic integrals of genus l.

Convexity of the Casimir map image on the energy surface is established.

Abstract

The C. Neumann system describes a particle on the sphere S^n under the influence of a potential that is a quadratic form. We study the case that the quadratic form has l+1 distinct eigenvalues with multiplicity. Each group of m_\sigma equal eigenvalues gives rise to an O(m_\sigma)-symmetry in configuration space. The combined symmetry group G is a direct product of l+1 such factors, and its cotangent lift has an Ad^*-equivariant Momentum mapping. Regular reduction leads to the Rosochatius system on S^l, which has the same form as the Neumann system albeit for an additional effective potential. To understand how the reduced systems fit together we use singular reduction to construct an embedding of the reduced Poisson space T^*{S^n}/G into R^{3l+3}$. The global geometry is described, in particular the bundle structure that appears as a result of the superintegrability of the system. We show how the reduced Neumann system separates in elliptical-spherical co-ordinates. We derive the action variables and frequencies as complete hyperelliptic integrals of genus l. Finally we prove a convexity result for the image of the Casimir mapping restricted to the energy surface.