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Marsden, Jerrold E.; Ratiu, Tudor

Reduction of Poisson manifolds. (English) Zbl 0602.58016

Lett. Math. Phys. 11, 161-169 (1986).
The authors introduce the notion of reduction in the category of Poisson manifolds. It is provided that a Poisson context is general enough to include the usual results on reduction of symplectic manifolds, Dirac brackets and the Lie-Poisson bracket. The functionality property of Poisson reduction is given. The dynamic counterpart of the Poisson reduction theorem is studied.
Some examples such as the Poisson reduction used by Arnol’d in passing from material to spatial coordinates in fluid dynamics and by Marsden and Weinstein for Vlasov equation, and the example closely related to the Hamiltonian structures used for the description of a particle in Yang- Mills field which gives an easy proof of the Adler-Kostant-Symes theorem, are regarded.
Reviewer: H.Kilp

Cited in 10 Reviews
Cited in 108 Documents

MSC:

53D20 Momentum maps; symplectic reduction
53D17 Poisson manifolds; Poisson groupoids and algebroids
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems

Keywords:

reduction in the category of Poisson manifolds; Dirac brackets; Lie- Poisson bracket; Poisson reduction; Yang-Mills field; Adler-Kostant-Symes theorem
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\textit{J. E. Marsden} and \textit{T. Ratiu}, Lett. Math. Phys. 11, 161--169 (1986; Zbl 0602.58016)
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References:

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