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

Holger Dullin; Heinz Hanßmann

Open Mathematics (2012)

  • Volume: 10, Issue: 5, page 1627-1654
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Holger Dullin, and Heinz Hanßmann. "The degenerate C. Neumann system I: symmetry reduction and convexity." Open Mathematics 10.5 (2012): 1627-1654. <http://eudml.org/doc/269160>.

@article{HolgerDullin2012,
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 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.},
author = {Holger Dullin, Heinz Hanßmann},
journal = {Open Mathematics},
keywords = {Superintegrable system; Symmetry reduction; Global action; Separating co-ordinates; Discriminant locus; superintegrable system; symmetry reduction; global action; separating co-ordinates; discriminant locus},
language = {eng},
number = {5},
pages = {1627-1654},
title = {The degenerate C. Neumann system I: symmetry reduction and convexity},
url = {http://eudml.org/doc/269160},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Holger Dullin
AU - Heinz Hanßmann
TI - The degenerate C. Neumann system I: symmetry reduction and convexity
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1627
EP - 1654
AB - 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.
LA - eng
KW - Superintegrable system; Symmetry reduction; Global action; Separating co-ordinates; Discriminant locus; superintegrable system; symmetry reduction; global action; separating co-ordinates; discriminant locus
UR - http://eudml.org/doc/269160
ER -

References

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