Research proof index

The corresponding magnetic field function from ℝ to Fin d × Fin d → ℝ of a plane wave. -/ noncomputable def magneticFunction {d : ℕ} {𝓕 : FreeSpace} {A : ElectromagneticPotential d} {s : Direction d} (hA : IsPlaneWave 𝓕 A s) : ℝ → Fin d × Fin d → ℝ := Classical.choose hA.2 lemma magneticFieldMatrix_eq_magneticFunction {d : ℕ} {𝓕 : FreeSpace} {A : E...

The kinetic energy from an electromagnetic potential. -/ noncomputable def kineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → ℝ := fun x => - 1/(4 * 𝓕.μ₀) * {η' d | μ μ' ⊗ η' d | ν ν' ⊗ A.toFieldStrength x | μ ν ⊗ A.toFieldStrength x | μ' ν'}ᵀ.toField /-! ### A.1. Lorentz invariance of the kinetic term We show that the kine...

The kinetic energy from an electromagnetic potential. -/ noncomputable def kineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → ℝ := fun x => - 1/(4 * 𝓕.μ₀) * {η' d | μ μ' ⊗ η' d | ν ν' ⊗ A.toFieldStrength x | μ ν ⊗ A.toFieldStrength x | μ' ν'}ᵀ.toField /-! ### A.1. Lorentz invariance of the kinetic term We show that the kine...

The kinetic energy from an electromagnetic potential. -/ noncomputable def kineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → ℝ := fun x => - 1/(4 * 𝓕.μ₀) * {η' d | μ μ' ⊗ η' d | ν ν' ⊗ A.toFieldStrength x | μ ν ⊗ A.toFieldStrength x | μ' ν'}ᵀ.toField /-! ### A.1. Lorentz invariance of the kinetic term We show that the kine...

The variational gradient of the kinetic term of an electromagnetic potential. -/ noncomputable def gradKineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → Lorentz.Vector d := (δ (q':=A), ∫ x, kineticTerm 𝓕 ⟨q'⟩ x) /-! ### B.1. Variational gradient in terms of fderiv We give a first simplification of the variational gradient...

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....