Smooth functions in the context of Poisson structure

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).