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.
, is naturally endowed with a binary operation called Poisson bracket, defined as
Symplectic geometry is therefore the natural mathematical setting to describe classical Hamiltonian mechanics.
This situation models the case of a physical system which is invariant under symmetries: the "reduced" phase space, obtained by quotienting the original phase space by the symmetries, in general is no longer symplectic, but is Poisson.
Although the modern definition of Poisson manifold appeared only in the 70's–80's,[1] its origin dates back to the nineteenth century.
Jacobi realized the importance of these brackets and elucidated their algebraic properties, and Lie began the study of their geometry.
Poisson computations occupied many pages, and his results were rediscovered and simplified two decades later by Carl Gustav Jacob Jacobi.
in order to reformulate (and give a much shorter proof of) Poisson's theorem on integrals of motion.
[16] The twentieth century saw the development of modern differential geometry, but only in 1977 André Lichnerowicz introduce Poisson structures as geometric objects on smooth manifolds.
[1] Poisson manifolds were further studied in the foundational 1983 paper of Alan Weinstein, where many basic structure theorems were first proved.
[17] These works exerted a huge influence in the subsequent decades on the development of Poisson geometry, which today is a field of its own, and at the same time is deeply entangled with many others, including non-commutative geometry, integrable systems, topological field theories and representation theory.
[15][10][11] There are two main points of view to define Poisson structures: it is customary and convenient to switch between them.
The class of vector spaces with linear Poisson structures coincides actually with that of (dual of) Lie algebras.
carries a linear Poisson bracket, known in the literature under the names of Lie-Poisson, Kirillov-Poisson or KKS (Kostant-Kirillov-Souriau) structure:
The class of vector bundles with linear Poisson structures coincides actually with that of (dual of) Lie algebroids.
The modular vector field of an orientable Poisson manifold, with respect to a volume form
An orientable Poisson manifold is called unimodular if its modular class vanishes.
For instance: The construction of the modular class can be easily extended to non-orientable manifolds by replacing volume forms with densities.
[34][37][38][24] A fundamental theorem states that the base space of any symplectic groupoid admits a unique Poisson structure
Roughly speaking, the role of a symplectic realisation is to "desingularise" a complicated (degenerate) Poisson manifold by passing to a bigger, but easier (nondegenerate), one.
In order to overcome this problem, one can use the notion of Poisson transversals (originally called cosymplectic submanifolds).
Several classes of Poisson manifolds have been shown to admit a canonical deformation quantisations:[47][48][49] In general, building a deformation quantisation for any given Poisson manifold is a highly non trivial problem, and for several years it was not clear if it would be even possible.
Alternative approaches and more direct constructions of Kontsevich's deformation quantisation were later provided by other authors.
[17][59] It is in general a difficult problem to determine if a given Poisson manifold is linearisable, and in many instances the answer is negative.
The original proof of Conn involves several estimates from analysis in order to apply the Nash-Moser theorem; a different proof, employing geometric methods which were not available at Conn's time, was provided by Crainic and Fernandes.
[63] If one restricts to analytic Poisson manifolds, a similar linearisation theorem holds, only requiring the isotropy Lie algebra
This was conjectured by Weinstein[17] and proved by Conn before his result in the smooth category;[64] a more geometric proof was given by Zung.
[65] Several other particular cases when the linearisation problem has a positive answer have been proved in the formal, smooth or analytic category.
This condition can be formulated in a number of equivalent ways:[66][67][68] It follows from the last characterisation that the Poisson bivector field
Accordingly, a non-trivial Poisson-Lie group cannot arise from a symplectic structure, otherwise it would contradict Weinstein splitting theorem applied to