Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The Poisson bracket satisfies the Jacobi identity

Statement

For all F,G,H∈C∞(M),

{F,{G,H}}+{G,{H,F}}+{H,{F,G}}=0.

Facts & Assumptions

Given: Three smooth functions on a symplectic manifold.

[F1]

The bracket is skew and [XF,XG]=−X{F,G}. The Poisson bracket is bilinear, skew, and a derivation in each entry, The Hamiltonian vector-field map is a Lie antihomomorphism.

Proof

technique · direct
1.1

Expand 0=dω(XF,XG,XH). Replacing derivatives by XA(B)={B,A} and commutators by [F1], the six terms combine in equal pairs to 0=2({{G,H},F}+{{H,F},G}+{{F,G},H}). This is a pointwise identity, not merely a statement that its differential vanishes.

F1givenalgebra
2.1

Divide by two and use skew-symmetry on each outer bracket. The result is the displayed Jacobi identity.

F1step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources