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Differential Structure of Orbit Spaces

  Published:2001-08-01
 Printed: Aug 2001
  • Richard Cushman
  • Jędrzej Śniatycki
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Abstract

We present a new approach to singular reduction of Hamiltonian systems with symmetries. The tools we use are the category of differential spaces of Sikorski and the Stefan-Sussmann theorem. The former is applied to analyze the differential structure of the spaces involved and the latter is used to prove that some of these spaces are smooth manifolds. Our main result is the identification of accessible sets of the generalized distribution spanned by the Hamiltonian vector fields of invariant functions with singular reduced spaces. We are also able to describe the differential structure of a singular reduced space corresponding to a coadjoint orbit which need not be locally closed.
Keywords: accessible sets, differential space, Poisson algebra, proper action, singular reduction, symplectic manifolds accessible sets, differential space, Poisson algebra, proper action, singular reduction, symplectic manifolds
MSC Classifications: 37J15, 58A40, 58D19, 70H33 show english descriptions Symmetries, invariants, invariant manifolds, momentum maps, reduction [See also 53D20]
Differential spaces
Group actions and symmetry properties
Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction
37J15 - Symmetries, invariants, invariant manifolds, momentum maps, reduction [See also 53D20]
58A40 - Differential spaces
58D19 - Group actions and symmetry properties
70H33 - Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction
 

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