False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Hamiltonian flows are complete on every symplectic manifold

Statement refuted

All Hamiltonian flows are complete.

Facts & Assumptions

Given: The proposed universal claim.

[F1]

Preservation of $ω$ is asserted only wherever the local Hamiltonian flow exists. Hamiltonian flows preserve the symplectic form.

[F2]

The convention $ι_{X_{H}} ω = d H$ defines the Hamiltonian field. Hamiltonian vector field and Hamiltonian function.

Refutation

technique · direct

1.1F2algebra

On $(R^{2}, d q \land d p)$ take $H (q, p) = q^{2} p$ . Writing $X_{H} = a \partial_{q} + b \partial_{p}$ , [F2] gives $a d p - b d q = d H = 2 q p d q + q^{2} d p$ , so $\dot{q} = a = q^{2}$ and $\dot{p} = b = - 2 q p$ . The solution from $(1, 0)$ has $p (t) = 0$ and $q (t) = 1 / (1 - t)$ for $t < 1$ .

2.1F1step 1.1∎

This trajectory escapes to infinity as $t ↑ 1$ and cannot be extended to a curve in $R^{2}$ at time one. Thus the smooth Hamiltonian field is incomplete; [F1] never claimed otherwise.

Depends on

Used by

Nothing in the library uses this result yet.

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

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)