Lagrangian has Var Gradient At eq add grad Kinetic Term
Electromagnetism.ElectromagneticPotential.lagrangian_hasVarGradientAt_eq_add_gradKineticTerm
Project documentation
The Lagrangian is equal to 1/2 * (ε₀ E^2 - 1/μ₀ B^2) - φρ + A · j-/ lemma lagrangian_eq_electric_magnetic {d} {𝓕 : FreeSpace} (A : ElectromagneticPotential d) (hA : ContDiff ℝ 2 A) (J : LorentzCurrentDensity d) (x : SpaceTime d) : A.lagrangian 𝓕 J x = 1 / 2 * (𝓕.ε₀ * ‖A.electricField 𝓕.c (x.time 𝓕.c) x.space‖ ^ 2 - (1 / (2 * 𝓕.μ₀)) * ∑ i, ∑ j, ‖A....
Exact Lean statement
lemma lagrangian_hasVarGradientAt_eq_add_gradKineticTerm {𝓕 : FreeSpace}
(A : ElectromagneticPotential d) (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d)
(hJ : ContDiff ℝ ∞ J) :
HasVarGradientAt (fun A => lagrangian 𝓕 ⟨A⟩ J)
(A.gradKineticTerm 𝓕 - A.gradFreeCurrentPotential J) AFormal artifact
Lean source
lemma lagrangian_hasVarGradientAt_eq_add_gradKineticTerm {𝓕 : FreeSpace} (A : ElectromagneticPotential d) (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d) (hJ : ContDiff ℝ ∞ J) : HasVarGradientAt (fun A => lagrangian 𝓕 ⟨A⟩ J) (A.gradKineticTerm 𝓕 - A.gradFreeCurrentPotential J) A := by conv => enter [1, q', x] rw [lagrangian] apply HasVarGradientAt.add · exact A.kineticTerm_hasVarGradientAt hA apply HasVarGradientAt.neg convert freeCurrentPotential_hasVarGradientAt A hA J hJ rw [← gradFreeCurrentPotential_eq_sum_basis A hA J hJ]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Electromagnetism/Dynamics/Lagrangian.lean:254-266
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