Poisson manifold

Abstract

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 M {\displaystyle M} is a function { ⋅ , ⋅ } : C ∞ ( M ) × C ∞ ( M ) → C ∞ ( M ) {\displaystyle \{\cdot ,\cdot \}:{\mathcal {C}}^{\infty }(M)\times {\mathcal {C}}^{\infty }(M)\to {\mathcal {C}}^{\infty }(M)}on the vector space C ∞ ( M ) {\displaystyle {\mathcal {C}}^{\infty }(M)} of smooth functions on M {\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[1] and are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics.[2]