Hamiltonian contact frame pairing

Proved

BirkhoffGlobalSection.hamiltonian_contact_frame_pairing

by Mazecto · Oct 3, 2026 · Mathlib 0df444a (Lean v4.33.1)

differential-geometryreal-analysis

The two projected Hamiltonian contact-frame vectors have the displayed symplectic pairing in terms of gradient radius.

All displayed regularity, tangency, normalization and supplied-data hypotheses of the exact formal statement are retained.

Preamble

import Theorems.Thm_BirkhoffGlobalSection_transverse_contact_frame_bundle
open BirkhoffGlobalSection Set Filter SignType
open scoped ContDiff Topology
set_option linter.unusedVariables false


set_option autoImplicit false
set_option maxHeartbeats 800000

Formal statement

theorem BirkhoffGlobalSection.hamiltonian_contact_frame_pairing :
∀ (F : Phase → ℝ) (y : Phase)
    (hp : 0 < phaseRadialContact y (hamiltonianVectorField F y)),
    phaseSymplecticForm (hamiltonianContactFrame F y 0) (hamiltonianContactFrame F y 1) =
      -(zNormSq (TangentialHessian.grad F y) + wNormSq (TangentialHessian.grad F y)) := by sorry

Source

Independent extraction of component 4 of the Proved theorem BirkhoffGlobalSection.transverse_contact_frame_bundle. Original verified source: Solutions/Sol_BirkhoffGlobalSection_transverse_contact_frame_bundle.lean. Exact original component retained.

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