Plane polar increment unique
ProvedBirkhoffGlobalSection.plane_polar_increment_uniquedynamical-systemstopology
Two continuous-angle, positive-radius polar representations of the same plane curve have identical angular increments relative to time zero.
All displayed regularity, tangency, normalization and supplied-data hypotheses of the exact formal statement are retained.
Preamble
import Theorems.Thm_BirkhoffGlobalSection_contact_angular_lift_bundle open BirkhoffGlobalSection Set Filter Topology open scoped ContDiff Topology set_option linter.unusedVariables false set_option autoImplicit false set_option maxHeartbeats 800000
Formal statement
theorem BirkhoffGlobalSection.plane_polar_increment_unique :
∀ (v : ℝ → Plane) (ρ σ θ ψ : ℝ → ℝ)
(hθ : Continuous θ) (hψ : Continuous ψ)
(hρ : ∀ t, 0 < ρ t) (hσ : ∀ t, 0 < σ t)
(hvθ : ∀ t, v t = ![ρ t * Real.cos (θ t), ρ t * Real.sin (θ t)])
(hvψ : ∀ t, v t = ![σ t * Real.cos (ψ t), σ t * Real.sin (ψ t)]),
∀ t, θ t - θ 0 = ψ t - ψ 0 := by sorrySource
Independent extraction of component 5 of the Proved theorem BirkhoffGlobalSection.contact_angular_lift_bundle. Original verified source: Solutions/Sol_BirkhoffGlobalSection_contact_angular_lift_bundle.lean. Exact original component retained.