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

• Autorzy: Holger Dullin , Heinz Hanßmann
• Języki publikacji: EN

Abstrakty

EN

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. 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 ℝ3ℓ+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.

Słowa kluczowe

EN

Superintegrable system; Symmetry reduction; Global action; Separating co-ordinates; Discriminant locus

Kategorie tematyczne

• 70H33: Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction
• 70H06: Completely integrable systems and methods of integration
• 37J35: Completely integrable systems, topological structure of phase space, integration methods
• 37J15: Symmetries, invariants, invariant manifolds, momentum maps, reduction
• 53D20: Momentum maps; symplectic reduction

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 5
• Strony: 1627-1654

Daty

wydano

2012-10-01

online

2012-07-24

Twórcy

autor

Holger Dullin

• University of Sydney

autor

Heinz Hanßmann

• Universiteit Utrecht

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Identyfikatory

DOI

10.2478/s11533-012-0085-8