Research proof index

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....

The magnetic field from the electromagnetic potential. -/ noncomputable def magneticField (c : SpeedOfLight := 1) (A : ElectromagneticPotential) : MagneticField := fun t x => (∇ ⨯ (A.vectorPotential c t)) x lemma magneticField_eq {c : SpeedOfLight} (A : ElectromagneticPotential) : A.magneticField c = fun t x => (∇ ⨯ (A.vectorPotential c t)) x := rfl /-! #...

The matrix corresponding to the magnetic field in general dimensions. In 3 space-dimensions this reduces to a vector. -/ noncomputable def magneticFieldMatrix (c : SpeedOfLight := 1) (A : ElectromagneticPotential d) : Time → Space d → (Fin d × Fin d) → ℝ := timeSlice c <| fun x ij => A.fieldStrengthMatrix x (Sum.inr ij.1, Sum.inr ij.2) lemma magneticFie...

The matrix corresponding to the magnetic field in general dimensions. In 3 space-dimensions this reduces to a vector. -/ noncomputable def magneticFieldMatrix (c : SpeedOfLight := 1) (A : ElectromagneticPotential d) : Time → Space d → (Fin d × Fin d) → ℝ := timeSlice c <| fun x ij => A.fieldStrengthMatrix x (Sum.inr ij.1, Sum.inr ij.2) lemma magneticFie...

The electric field of the electromagnetic potential created from the electric field E and the magnetic field B is E, as long as Gauss's law for magnetism and Faraday's law are satisfied.