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

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 _σ equal eigenvalues gives rise to an O(m _σ )-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.