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.
Read more about this topic: Poisson Manifold
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