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

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Spherical-pendulum monodromy obstructs global action–angle coordinates

Example

This item assumes ${A C}_{ω}$ , namely countable choice. In the propagated dependency chain, that assumption is required through Period-lattice monodromy obstructs global action–angle coordinates; after those interfaces are fixed, the remaining local or finite argument makes no additional countable-family choice.

Identify covectors and tangent vectors on the unit sphere by its round metric. The spherical pendulum has integral map

$F = (H, J) : T^{*} S^{2} ⟶ R^{2}, H = \frac{1}{2} ∣ p ∣^{2} + z, J = x p_{y} - y p_{x} .$

The regular torus fibration has nontrivial monodromy around the focus–focus value $(1, 0)$ and therefore has no global action–angle chart on that punctured regular region.

Facts & Assumptions

Given: ${A C}_{ω}$ , the source's orientation of the loop and its homology basis $(a, b)$ , where $b$ is an orbit of the circle action generated by $J$ .

[A1]

${A C}_{ω}$ is countable choice and is required here through Period-lattice monodromy obstructs global action–angle coordinates; after those supplied interfaces are fixed, the remaining local or finite calculation makes no additional countable-family choice.

[F1]

Nonidentity period-lattice monodromy obstructs global action–angle coordinates. Period-lattice monodromy obstructs global action–angle coordinates.

[F2]

Martynchuk–Broer–Efstathiou, Theorem 2.7 and §3.1, prove that crossing the unique focus–focus critical energy $h = 1$ changes the energy-level Chern number by $c (1 + ε) - c (1 - ε) = 1$ and compute the associated gluing matrices.

Verification

Proof technique: direct, with the advanced index computation imported from the cited source.

1.1

The Hamiltonian flow of $J$ is the axial circle action, so its orbit cycle $b$ is unchanged by transport around the source's positively oriented loop $γ$ enclosing $(1, 0)$ .

given

1.2

By [F2], the two energy-level pieces used to cross the critical energy have Chern numbers $c_{1} = c_{2} + 1$ . The source's solid-torus gluing sends the basis by $T (c) = (\begin{array}{cc} 1 & c \\ 0 & 1 \end{array})$ ; hence its complete calculation gives $M_{γ} = T (c_{1}) T (c_{2})^{- 1} = (\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array})$ in the basis $(a, b)$ . This step invokes, rather than reproves, the source's Takens index theorem and gluing argument.

2.1

The matrix is not the identity, so [F1] rules out a single global action–angle chart on the regular torus bundle around the puncture. Reversing the loop or changing the integral basis can invert or conjugate the matrix but cannot make its monodromy trivial.

A1F1step 1.2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Nikolay Martynchuk, Henk W. Broer, and Konstantinos Efstathiou, Hamiltonian Monodromy and Morse Theory (standard reference, not scraped)