Research proof index

The canonical momentum associated with the lagrangian of an electromagnetic potential and a Lorentz current density. -/ noncomputable def canonicalMomentum (𝓕 : FreeSpace) (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) : SpaceTime d → Lorentz.Vector d := fun x => gradient (fun (v : Lorentz.Vector d) => lagrangian 𝓕 ⟨fun x => A x + x (Sum...

The canonical momentum associated with the lagrangian of an electromagnetic potential and a Lorentz current density. -/ noncomputable def canonicalMomentum (𝓕 : FreeSpace) (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) : SpaceTime d → Lorentz.Vector d := fun x => gradient (fun (v : Lorentz.Vector d) => lagrangian 𝓕 ⟨fun x => A x + x (Sum...

An electric potential which gives a given constant E-field and B-field. -/ @[nolint unusedArguments] noncomputable def constantEB {d : ℕ} (c : SpeedOfLight) (E₀ : EuclideanSpace ℝ (Fin d)) (B₀ : Fin d × Fin d → ℝ) (B₀_antisymm : ∀ i j, B₀ (i, j) = - B₀ (j, i)) : ElectromagneticPotential d where val := fun x μ => match μ with | Sum.inl _ => - (1/c) * ⟪E₀,...

An electric potential which gives a given constant E-field and B-field. -/ @[nolint unusedArguments] noncomputable def constantEB {d : ℕ} (c : SpeedOfLight) (E₀ : EuclideanSpace ℝ (Fin d)) (B₀ : Fin d × Fin d → ℝ) (B₀_antisymm : ∀ i j, B₀ (i, j) = - B₀ (j, i)) : ElectromagneticPotential d where val := fun x μ => match μ with | Sum.inl _ => - (1/c) * ⟪E₀,...

An electric potential which gives a given constant E-field and B-field. -/ @[nolint unusedArguments] noncomputable def constantEB {d : ℕ} (c : SpeedOfLight) (E₀ : EuclideanSpace ℝ (Fin d)) (B₀ : Fin d × Fin d → ℝ) (B₀_antisymm : ∀ i j, B₀ (i, j) = - B₀ (j, i)) : ElectromagneticPotential d where val := fun x μ => match μ with | Sum.inl _ => - (1/c) * ⟪E₀,...

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...