Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Liouville volume preservation

Statement

On a 2n-dimensional symplectic manifold, every Hamiltonian local flow preserves the Liouville volume form Ω=ωn/n!.

Facts & Assumptions

Given: A Hamiltonian vector field and its local flow ϕt.

[F1]

The symplectic volume is Ω=ωn/n!. Symplectic manifolds have a canonical orientation and volume form.

[F2]

Hamiltonian local flows satisfy ϕt∗ω=ω. Hamiltonian flows preserve the symplectic form.

Proof

technique · direct
1.1

Pullback respects wedges and scalar multiplication, so [F2] gives ϕt∗Ω=(ϕt∗ω)n/n!=ωn/n!.

F1F2given
2.1

Thus ϕt∗Ω=Ω wherever the local flow exists. This is volume preservation, with no completeness conclusion.

F1step 1.1∎

Depends on

Used by

Dependency tree · two levels

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Sources