Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

One Dim Point Particle scalar Potential

Electromagnetism.DistElectromagneticPotential.oneDimPointParticle_scalarPotential

physics quantum field theory relativity one dimension

Project documentation

The electromagnetic potential of a point particle stationary at r₀ of 1d space. -/ noncomputable def oneDimPointParticle (𝓕 : FreeSpace) (q : ℝ) (r₀ : Space 1) : DistElectromagneticPotential 1 := (SpaceTime.distTimeSlice 𝓕.c).symm <| Space.constantTime <| distOfFunction (fun x => ((- (q * 𝓕.μ₀ * 𝓕.c)/ 2) * ‖x - r₀‖) • Lorentz.Vector.basis (Sum.inl 0...

Exact Lean statement

lemma oneDimPointParticle_scalarPotential (𝓕 : FreeSpace) (q : ℝ) (r₀ : Space 1) :
    (oneDimPointParticle 𝓕 q r₀).scalarPotential 𝓕.c =
    Space.constantTime (distOfFunction (fun x =>
      - ((q * 𝓕.μ₀ * 𝓕.c ^ 2)/(2)) • ‖x-r₀‖) (by fun_prop))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma oneDimPointParticle_scalarPotential (𝓕 : FreeSpace) (q : ℝ) (r₀ : Space 1) :    (oneDimPointParticle 𝓕 q r₀).scalarPotential 𝓕.c =    Space.constantTime (distOfFunction (fun x =>      - ((q * 𝓕.μ₀ * 𝓕.c ^ 2)/(2)) • ‖x-r₀‖) (by fun_prop)) := by  ext ε  simp only [scalarPotential, Lorentz.Vector.temporalCLM, Fin.isValue, map_smul,    ContinuousLinearMap.comp_smulₛₗ, Real.ringHom_apply, oneDimPointParticle, LinearMap.coe_mk,    AddHom.coe_mk, ContinuousLinearEquiv.apply_symm_apply, FunLike.coe_smul,    ContinuousLinearMap.coe_comp, LinearMap.coe_toContinuousLinearMap', Pi.smul_apply,    Function.comp_apply, constantTime_apply, distOfFunction_vector_eval, Lorentz.Vector.apply_smul,    Lorentz.Vector.basis_apply, ↓reduceIte, mul_one, smul_eq_mul, neg_mul]  rw [distOfFunction_mul_fun _ (by fun_prop), distOfFunction_neg,    distOfFunction_mul_fun _ (by fun_prop)]  simp only [FunLike.coe_smul, Pi.smul_apply, smul_eq_mul,    _root_.neg_apply]  ring

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/Electromagnetism/PointParticle/OneDimension.lean:143-158

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.32.0

Adiabatic relation log

adiabatic_relation_log

Plain-language statement

Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Adiabatic relation Ua Ub Va Vb

adiabatic_relation_UaUbVaVb

Plain-language statement

Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Deriv Within mean Energy Beta eq neg variance

CanonicalEnsemble.derivWithin_meanEnergy_Beta_eq_neg_variance

Plain-language statement

(∂U/∂β) = -Var(E) for finite systems.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record