LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

A smooth local Hamiltonian flow preserves the canonical symplectic matrix

Statement

Let $U \subseteq R^{2 d}$ be open, $H \in C^{\infty} (U)$ , and use the canonical symplectic form $ω = \sum_{i = 1}^{d} d q_{i} \land d p_{i}$ with matrix $J = (\begin{array}{cc} 0 & I \\ - I & 0 \end{array})$ . Define $X_{H}$ by $ι_{X_{H}} ω = d H$ . Let $φ_{t}$ be the local flow on any open set where it is defined. At each initial state and time for which the intervening trajectory lies in $U$ , its derivative satisfies $D φ_{t} (z)^{T} J D φ_{t} (z) = J .$ This is a local, choice-free variational proof; no completeness is asserted.

Facts & Assumptions

Given: The open set, smooth Hamiltonian, canonical matrix, and local Hamiltonian flow.

[F1]

The library convention is $ι_{X_{H}} ω = d H$ . In these coordinates, if $X = (u, v)$ then $ι_{X} ω = \sum_{i} (u_{i} d p_{i} - v_{i} d q_{i})$ ; equating this with $d H$ gives $X_{H} = (\nabla_{p} H, - \nabla_{q} H) = J \nabla H$ . Hamiltonian vector field and Hamiltonian function

[F2]

The local flow is $C^{1}$ in its initial state and its derivative solves $Y^{'} = D X_{H} (φ_{t} (z)) Y$ , $Y (0) = I$ . C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade

[F3]

The Hessian of a smooth scalar function is symmetric. Continuous mixed partials of order $k$ are invariant under permutations

Proof

technique · differentiate the pulled-back constant matrix along the variational equation

1.1F1F3algebra

In canonical coordinates [F1] gives $X_{H} = J \nabla H$ , hence $A (z) : = D X_{H} (z) = J \nabla^{2} H (z)$ . By [F3] the Hessian is symmetric, and $J^{T} = - J$ , $J^{2} = - I$ ; therefore $A^{T} J + J A = 0$ .

2.1F2step 1.1algebra∎

Set $Y (t) = D φ_{t} (z)$ . By [F2], $Y^{'} = A (φ_{t} (z)) Y$ and $Y (0) = I$ . The product rule gives $\frac{d}{d t} (Y^{T} J Y) = Y^{T} (A^{T} J + J A) Y = 0$ by step 1.1. Thus $Y (t)^{T} J Y (t) = Y (0)^{T} J Y (0) = J$ throughout the local time interval.

Depends on

Used by

A finite formal logarithm has Hamiltonian coefficients on a star-shaped domain Lemma

Dependency tree · two levels

17 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

Ernst Hairer, Geometric Numerical Integration, TU München Lecture 1: Hamiltonian systems (standard reference, not scraped)
Nikolaj Nordkvist and Poul G. Hjorth, Classical Mechanics and Symplectic Integration (standard reference, not scraped)