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Advanced MechanicsFrom Euler's Determinism to Arnold's Chaos$
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S. G. Rajeev

Print publication date: 2013

Print ISBN-13: 9780199670857

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199670857.001.0001

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Poisson and symplectic manifolds

Poisson and symplectic manifolds

(p.111) 14 Poisson and symplectic manifolds
Advanced Mechanics

S. G. Rajeev

Oxford University Press

The Poisson bracket defines a tensor on the phase space which satisfies an integrability condition. When this tensor is invertible, this condition is linear and gives a symplectic structure. Liouville's theorem is proved. The symplectic geometry of the sphere is explained, and shown to give a classical description of spin. Although classical, spin cannot be described by canonical variables valid in the whole phase space: a sphere cannot be covered by any single co-ordinate chart. An application to nuclear magnetic resonance is described. An exercise introduces the Grassmannian manifolds and Poisson brackets on them.

Keywords:   Grassmannian manifolds, symplectic structure, Liouville's theorem, spin, nuclear magnetic resonance

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