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

How statement and proof provenance work

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.

Geodesic flow as a Hamiltonian flow on the cotangent bundle

Example

Assume ${A C}_{ω}$ . For a supplied Riemannian manifold $(Q, g)$ , the Hamiltonian

$H (q, p) = \frac{1}{2} g_{q}^{- 1} (p, p)$

on $T^{*} Q$ generates geodesic flow: under $g^{♭} : T Q \to T^{*} Q$ , its Hamiltonian trajectories are precisely the tangent lifts $(q (t), \dot{q} (t))$ of affinely parametrized geodesics.

Facts & Assumptions

Given: A smooth Riemannian metric $g$ .

[F1]

The metric has a unique Levi–Civita connection. Fundamental theorem of riemannian geometry.

[F2]

The natural Lagrangian with zero potential has Legendre map $g^{♭}$ and the displayed Hamiltonian. A natural mechanical Lagrangian gives the kinetic-plus-potential Hamiltonian.

[F3]

Under the stated choice assumption, the Legendre map bijects Euler–Lagrange and Hamiltonian trajectories. Equivalence of Euler–Lagrange and Hamilton equations for hyperregular Lagrangians.

Verification

technique · direct

1.1

For $L (q, v) = \frac{1}{2} g_{q} (v, v)$ , [F2] gives $F L = g^{♭}$ and $H = \frac{1}{2} g^{- 1} (p, p)$ . The Euler–Lagrange equation of this kinetic-energy Lagrangian is the coordinate equation ${\ddot{q}}^{k} + Γ_{i j}^{k} {\dot{q}}^{i} {\dot{q}}^{j} = 0$ for the Levi–Civita connection in [F1], hence is $\nabla_{\dot{q}} \dot{q} = 0$ .

F1F2algebra

2.1

By [F3], $(q, \dot{q})$ solves that geodesic equation exactly when $(q, p) = (q, g^{♭} \dot{q})$ is a Hamiltonian trajectory of $H$ . Thus the two flows correspond wherever their maximal trajectories exist; no completeness of $g$ is asserted.

F3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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)