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

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A natural mechanical Lagrangian gives the kinetic-plus-potential Hamiltonian

Statement

For a Riemannian metric $g$ and potential $V$ , the natural Lagrangian

$L (q, v) = \frac{1}{2} g_{q} (v, v) - V (q)$

is hyperregular, with $F L = g^{♭}$ , and its Hamiltonian is

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

Facts & Assumptions

Given: A smooth Riemannian metric $g$ and smooth potential $V$ .

[F1]

For hyperregular $L$ , $E_{L} = p (v) - L$ and $H = E_{L} \circ (F L)^{- 1}$ . Energy and Hamiltonian of a hyperregular Lagrangian.

Proof

technique · direct

1.1

Fibre differentiation gives $F L (q, v) = g_{q} (v, \cdot) = g_{q}^{♭} (v)$ . Positive definiteness makes $g^{♭}$ a smooth bundle isomorphism with inverse $g^{♯}$ , so $L$ is hyperregular.

givenalgebra

2.1

With $p = g^{♭} v$ , [F1] gives $E_{L} = g (v, v) - \frac{1}{2} g (v, v) + V = \frac{1}{2} g (v, v) + V$ . Substituting $v = g^{♯} p$ yields $H (q, p) = \frac{1}{2} g^{- 1} (p, p) + V (q)$ .

F1step 1.1algebra∎

Depends on

Energy and Hamiltonian of a hyperregular Lagrangian

Used by

Dependency tree · two levels

2 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)