Poisson Manifold - Definition

Definition

A Poisson bracket (or Poisson structure) on a smooth manifold M is a bilinear map

that satisfies the following three properties:

  • It is skew-symmetric: {f,g} = - {g,f}.
  • It obeys the Jacobi Identity: {f,{g,h}} + {g,{h,f}} + {h,{f,g}} = 0.
  • It obeys Leibniz's Rule with respect to the first argument: {fg,h} = f{g,h} + g{f,h}.

By skew-symmetry, the Poisson bracket automatically satisfies Leibniz's Rule with respect to the second argument. The last property basically states that the map f ↦ {f,g} is a derivation on C∞(M) for any fixed g ∈ C∞(M). Every derivation δ on C∞(M) can be written as a directional derivative (x) = (df)x((Xδ)x), where x ∈ M, for some vector field Xδ. It follows that for g ∈ C∞(M), we obtain a vector field Xg such that {f,g}(x) = (df)x((Xg)x), where x ∈ M (written more briefly, {f,g} = df(Xg)). The vector field Xg is called the Hamiltonian vector field corresponding to g. Notice that

,

where <⋅,⋅> is the pairing between the cotangent and tangent bundles of M. Therefore, {f,g} depends only on the differentials df and dg. Any Poisson bracket yields a map from the cotangent bundle to the tangent bundle that sends df to Xf.

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