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

100%

Dullin H., Hanßmann H.

Open Mathematics

2012

tom 10

nr 5

1627-1654

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.

The gap phenomenon in the dimension study of finite type systems

100%

Kruglikov B.

Open Mathematics

2012

tom 10

nr 5

1605-1618

Several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result clarifying this effect in the case when the structure is associated to a vector distribution, is proposed.

Ograniczanie wyników

2 Open Mathematics

1 Dullin H.

1 Hanßmann H.

1 Kruglikov B.

2 2012

Wyniki wyszukiwania

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

The gap phenomenon in the dimension study of finite type systems