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Is Extrema iff gauss ampere magnetic Field Matrix

Electromagnetism.ElectromagneticPotential.isExtrema_iff_gauss_ampere_magneticFieldMatrix

physics quantum field theory relativity is extrema

Project documentation

The condition on an electromagnetic potential to be an extrema of the lagrangian. -/ def IsExtrema {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) : Prop := gradLagrangian 𝓕 A J = 0 lemma isExtrema_iff_gradLagrangian {𝓕 : FreeSpace} (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) : IsExtrema 𝓕 A J ↔ A....

Exact Lean statement

lemma isExtrema_iff_gauss_ampere_magneticFieldMatrix {d} {𝓕 : FreeSpace}
    {A : ElectromagneticPotential d}
    (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d)
    (hJ : ContDiff ℝ ∞ J) :
    IsExtrema 𝓕 A J ↔ ∀ t, ∀ x, (∇ ⬝ (A.electricField 𝓕.c t)) x = J.chargeDensity 𝓕.c t x / 𝓕.ε₀
    ∧ ∀ i, 𝓕.μ₀ * 𝓕.ε₀ * ∂ₜ (fun t => A.electricField 𝓕.c t x) t i =
    ∑ j, ∂[j] (A.magneticFieldMatrix 𝓕.c t · (j, i)) x - 𝓕.μ₀ * J.currentDensity 𝓕.c t x i

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma isExtrema_iff_gauss_ampere_magneticFieldMatrix {d} {𝓕 : FreeSpace}    {A : ElectromagneticPotential d}    (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d)    (hJ : ContDiff ℝ ∞ J) :    IsExtrema 𝓕 A J ↔ ∀ t, ∀ x, (∇ ⬝ (A.electricField 𝓕.c t)) x = J.chargeDensity 𝓕.c t x / 𝓕.ε₀    ∧ ∀ i, 𝓕.μ₀ * 𝓕.ε₀ * ∂ₜ (fun t => A.electricField 𝓕.c t x) t i =    ∑ j, ∂[j] (A.magneticFieldMatrix 𝓕.c t · (j, i)) x - 𝓕.μ₀ * J.currentDensity 𝓕.c t x i := by  rw [isExtrema_iff_gradLagrangian]  rw [funext_iff]  conv_lhs =>    enter [x]    rw [gradLagrangian_eq_electricField_magneticField (𝓕 := 𝓕) A hA J hJ]    simp only [Pi.zero_apply]    rw [Lorentz.Vector.sum_inl_inr_basis_eq_zero_iff]  simp only [forall_and]  apply and_congr  · apply Iff.intro    · intro h t x      specialize h ((toTimeAndSpace 𝓕.c).symm (t, x))      simp at h      linear_combination (norm := simp) (𝓕.μ₀ * 𝓕.c) * h      field_simp      simp only [FreeSpace.c_sq, one_div, mul_inv_rev, mul_zero]      field_simp      ring    · intro h x      specialize h (x.time 𝓕.c) x.space      linear_combination (norm := simp) (𝓕.μ₀⁻¹ * 𝓕.c⁻¹) * h      field_simp      simp only [FreeSpace.c_sq, one_div, mul_inv_rev, mul_zero]      field_simp      ring  · apply Iff.intro    · intro h t x i      specialize h ((toTimeAndSpace 𝓕.c).symm (t, x)) i      simp at h      linear_combination (norm := simp) (𝓕.μ₀) * h      field_simp      simp    · intro h x i      specialize h (x.time 𝓕.c) x.space i      linear_combination (norm := simp) (𝓕.μ₀⁻¹) * h      field_simp      simp

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/Electromagnetism/Dynamics/IsExtrema.lean:229-272

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