Abstract
In this chapter we prove, among other things, that the coadjoint orbits of a Lie group are symplectic manifolds. These symplectic manifolds are, in fact, the symplectic leaves for the Lie-Poisson bracket. This result was developed and used by Kirillov, Arnold, Kostant, and Souriau in the early to mid 1960s, although it had important roots going back to the work of Lie, Borel, and Weil. (See Kirillov [1962, 1976b], Arnold [1966a], Kostant [1970], and Souriau [1970].) Here we give a direct proof. Alternatively, one can give a proof using general reduction theory, as in Marsden and Weinstein [1974] and Abraham and Marsden [1978].
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© 1999 Springer Science+Business Media New York
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Marsden, J.E., Ratiu, T.S. (1999). Coadjoint Orbits. In: Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, vol 17. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21792-5_14
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DOI: https://doi.org/10.1007/978-0-387-21792-5_14
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-3143-6
Online ISBN: 978-0-387-21792-5
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