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

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Hamilton equations in canonical cotangent coordinates

Statement

Assume ${A C}_{ω}$ . In canonical coordinates on $T^{*} Q$ , an integral curve $(q (t), p (t))$ of $X_{H}$ satisfies

${\dot{q}}^{i} = \frac{\partial H}{\partial p_{i}}, {\dot{p}}_{i} = - \frac{\partial H}{\partial q^{i}} .$

Facts & Assumptions

Given: The cotangent convention $ω = \sum_{i} d q^{i} \land d p_{i}$ and $ι_{X_{H}} ω = d H$ .

[F1]

The canonical cotangent form has the displayed coordinate expression. The canonical cotangent two-form is symplectic.

[F2]

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

Proof

technique · direct

1.1F1givenalgebra

Write $X_{H} = \sum_{i} (a^{i} \partial_{q^{i}} + b_{i} \partial_{p_{i}})$ . Then [F1] gives $ι_{X_{H}} ω = \sum_{i} (a^{i} d p_{i} - b_{i} d q^{i})$ .

2.1F2step 1.1∎

Comparing with $d H = \sum_{i} (H_{q^{i}} d q^{i} + H_{p_{i}} d p_{i})$ in [F2] gives $a^{i} = H_{p_{i}}$ and $b_{i} = - H_{q^{i}}$ . Since an integral curve has velocity $X_{H}$ , these are Hamilton's equations.

Depends on

Used by

Free particle Hamiltonian flow Example
Harmonic oscillator and elliptic phase curves Example
Simple pendulum phase portrait Example
Coordinate formula for the Poisson bracket Proposition
Equivalence of Euler–Lagrange and Hamilton equations for hyperregular Lagrangians Theorem

Dependency tree · two levels

10 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)