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 ${\displaystyle M}$ is a function$${\displaystyle \{\cdot ,\cdot \}:{\mathcal {C}}^{\infty }(M)\times {\mathcal {C}}^{\infty }(M)\to {\mathcal {C}}^{\infty }(M)}$$on the vector space ${\displaystyle {\mathcal {C}}^{\infty }(M)}$ of smooth functions on ${\displaystyle M}$, 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.

Poisson geometry can be regarded as a combination of foliation theory, symplectic geometry, and Lie theory. A Poisson manifold foliates. Each leaf of the foliation has a symplectic structure. The leaves are connected transversely through Lie geometry.

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 ${\displaystyle n}$-dimensional Euclidean space (i.e. having ${\displaystyle \mathbb {R} ^{n}}$ as configuration space) has phase space ${\displaystyle \mathbb {R} ^{2n}}$. The coordinates ${\displaystyle (q^{1},...,q^{n},p_{1},...,p_{n})}$ describe respectively the positions and the generalised momenta. The space of observables, i.e. the smooth functions on ${\displaystyle \mathbb {R} ^{2n}}$, is naturally endowed with a binary operation called Poisson bracket, defined as ${\displaystyle \textstyle \{f,g\}:=\sum _{i=1}^{n}\left({\frac {\partial f}{\partial p_{i}}}{\frac {\partial g}{\partial q^{i}}}-{\frac {\partial f}{\partial q^{i}}}{\frac {\partial g}{\partial p_{i}}}\right)}$. Such bracket satisfies the standard properties of a Lie bracket, plus a further compatibility with the product of functions, namely the Leibniz identity ${\displaystyle \{f,g\cdot h\}=g\cdot \{f,h\}+\{f,g\}\cdot h}$. Equivalently, the Poisson bracket on ${\displaystyle \mathbb {R} ^{2n}}$ can be reformulated using the symplectic form ${\displaystyle \textstyle \omega :=\sum _{i=1}^{n}dq^{i}\wedge dp_{i}}$. Indeed, if one considers the Hamiltonian vector field ${\displaystyle \textstyle X_{f}:=\sum _{i=1}^{n}\left({\frac {\partial f}{\partial p_{i}}}\partial _{q^{i}}-{\frac {\partial f}{\partial q^{i}}}\partial _{p_{i}}\right)}$ associated to a function ${\displaystyle f}$, then the Poisson bracket can be rewritten as ${\displaystyle \{f,g\}=\omega (X_{g},X_{f}).}$

In more abstract differential geometric terms, the configuration space is an ${\displaystyle n}$-dimensional smooth manifold ${\displaystyle Q}$, and the phase space is its cotangent bundle ${\displaystyle T^{*}Q}$ (a manifold of dimension ${\displaystyle 2n}$). The latter is naturally equipped with a canonical symplectic form, which in canonical coordinates coincides with the one described above. In general, by Darboux theorem, any arbitrary symplectic manifold ${\displaystyle (M,\omega )}$ admits special coordinates where the form ${\displaystyle \omega }$ and the bracket ${\displaystyle \{f,g\}=\omega (X_{g},X_{f})}$ are equivalent with, respectively, the symplectic form and the Poisson bracket of ${\displaystyle \mathbb {R} ^{2n}}$. Symplectic geometry is therefore the natural mathematical setting to describe classical Hamiltonian mechanics.

Poisson manifolds are further generalisations of symplectic manifolds, which arise by axiomatising the properties satisfied by the Poisson bracket on ${\displaystyle \mathbb {R} ^{2n}}$. More precisely, a Poisson manifold consists of a smooth manifold ${\displaystyle M}$ (not necessarily of even dimension) together with an abstract bracket ${\displaystyle \{\cdot ,\cdot \}:{\mathcal {C}}^{\infty }(M)\times {\mathcal {C}}^{\infty }(M)\to {\mathcal {C}}^{\infty }(M)}$, still called Poisson bracket, which does not necessarily arise from a symplectic form ${\displaystyle \omega }$, but satisfies the same algebraic properties.