Mdifferentiable At iff differentiable At coord
ClassicalMechanics.HarmonicOscillator.Trajectory.mdifferentiableAt_iff_differentiableAt_coord
Plain-language statement
A trajectory is MDifferentiableAt a time iff its coordinate curve is ordinarily differentiable there, since the two differ by the coordinate diffeomorphism.
Exact Lean statement
lemma mdifferentiableAt_iff_differentiableAt_coord (γ : Trajectory) (t : Time) :
MDifferentiableAt 𝓘(ℝ, Time) 𝓘(ℝ, EuclideanSpace ℝ (Fin 1)) γ t ↔
DifferentiableAt ℝ (coord γ) tFormal artifact
Lean source
lemma mdifferentiableAt_iff_differentiableAt_coord (γ : Trajectory) (t : Time) : MDifferentiableAt 𝓘(ℝ, Time) 𝓘(ℝ, EuclideanSpace ℝ (Fin 1)) γ t ↔ DifferentiableAt ℝ (coord γ) t := by rw [← mdifferentiableAt_iff_differentiableAt] constructor · intro hγ rw [show coord γ = ConfigurationSpace.valDiffeomorph ∘ γ by rfl] exact MDifferentiableAt.comp t ((ConfigurationSpace.valDiffeomorph.mdifferentiable WithTop.top_ne_zero).mdifferentiableAt) hγ · intro hcoord have hγ : MDifferentiableAt 𝓘(ℝ, Time) 𝓘(ℝ, EuclideanSpace ℝ (Fin 1)) (ConfigurationSpace.valDiffeomorph.symm ∘ coord γ) t := MDifferentiableAt.comp t ((ConfigurationSpace.valDiffeomorph.symm.mdifferentiable WithTop.top_ne_zero).mdifferentiableAt) hcoord convert hγ using 1 funext s exact (ConfigurationSpace.valDiffeomorph.symm_apply_apply (γ s)).symm- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/ClassicalMechanics/HarmonicOscillator/Geometric/Trajectory.lean:112-130
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