Poisson and symplectic manifolds
Poisson and symplectic manifolds
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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