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Published 2000 | Submitted
Book Section - Chapter Open

Poisson structure and invariant manifolds on Lie groups

Abstract

For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian ℓ we define a Poisson structure via the pull-back of the Lie-Poisson structure on g^∗ by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group G of the canonical discrete Lagrange 2-form ω_L on G × G. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system.

Additional Information

© World Scientific, 2000. The authors would like to thank Alan Weinstein for pointing out the connections with the general theory of dynamics on groupoids and algebroids.

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