functions and the (Lie) bracket of their associated Hamiltonian vector fields, i.e.in order to reformulate (and give a much shorter proof of) Poisson's theorem on integrals of motion. Jacobi’s work on Poisson brackets inﬂuenced the pioneering studies of Sophus Lie on symmetries of diﬀerential equations, which led to the discovery of Lie groups and Lie algebras. For instance, what are now called linear Poisson structures (i.e. Poisson brackets on a vector space which send linear functions to linear functions) correspond precisely to Lie algebra structures. Moreover, the integrability of a linear Poisson structure (see below) is closely related to the integrability of its associated Lie algebra to a Lie group. The twentieth century saw the development of modern differential geometry, but only in 1977 André Lichnerowicz introduced Poisson structures as geometric objects on smooth manifolds. Poisson manifolds were further studied in the foundational 1983 paper of Alan Weinstein, where many basic structure theorems were first proved. 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 e.g. non-commutative geometry, integrable systems, topological field theories and representation theory. Formal definition There are two main points of view to define Poisson structures: it is customary and convenient to switch between them. As bracket Let be a smooth manifold and let denote the real algebra of smooth real-valued functions on , where the multiplication is defined pointwise. A Poisson bracket (or Poisson structure) on is an -bilinear map defining a structure of Poisson algebra on , i.e. satisfying the following three conditions: Skew symmetry: . Jacobi identity: . Leibniz's Rule: . The first two conditions ensure that defines a Lie-algebra structure on , while the third guarantees that, for each , the linear map is a derivation of the algebra , i.e., it defines a vector field called the Hamiltonian vector field associated to . Choosing local coordinates , any Poisson bracket is given byfor the Poisson bracket of the coordinate functions. As bivector A Poisson bivector on a smooth manifold is a bivector field satisfying the non-linear partial differential equation , where denotes the Schouten–Nijenhuis bracket on multivector fields. Choosing local coordinates , any Poisson bivector is given byfor skew-symmetric smooth functions on . Equivalence of the definitions Let be a bilinear skew-symmetric bracket (also called an almost Lie