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Concept
Poisson manifold
Summary
In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.
A Poisson structure (or Poisson bracket) on a smooth manifold is a functionon the vector space of smooth functions on , making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra).
Poisson structures on manifolds were introduced by André Lichnerowicz in 1977 and are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics.
In classical mechanics, the phase space of a physical system consists of all the possible values of the position and of the momentum variables allowed by the system. It is naturally endowed with a Poisson bracket/symplectic form (see below), which allows one to formulate the Hamilton equations and describe the dynamics of the system through the phase space in time.
For instance, a single particle freely moving in the -dimensional Euclidean space (i.e. having as configuration space) has phase space . The coordinates describe respectively the positions and the generalised momenta. The space of observables, i.e. the smooth functions on , is naturally endowed with a binary operation called Poisson bracket, defined as . Such bracket satisfies the standard properties of a Lie bracket, plus a further compatibility with the product of functions, namely the Leibniz identity . Equivalently, the Poisson bracket on can be reformulated using the symplectic form . Indeed, if one considers the Hamiltonian vector field associated to a function , then the Poisson bracket can be rewritten as
In more abstract differential geometric terms, the configuration space is an -dimensional smooth manifold , and the phase space is its cotangent bundle (a manifold of dimension ). The latter is naturally equipped with a canonical symplectic form, which in canonical coordinates coincides with the one described above.
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