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The degenerate C. Neumann system I: symmetry reduction and convexity | | Holger R. Dullin
; Heinz Hanßmann
; | | Date: |
8 May 2012 | | 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. | | Source: | arXiv, 1205.1834 | | Services: | Forum | Review | PDF | Favorites | |
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