Relativistic prescribed-field particle action yields the Lorentz force

Given: A nondegenerate compact interval $[a,b]$; a $C^2$ future timelike massive worldline $X:[a,b]\to A$ with four-velocity $U$ and proper time $\tau$; a prescribed $C^2$ covector field $A_{\mu }$ on a neighbourhood of the image of $X$; the rest-mass and charge parameters $m>0$, $q\in \mathbb{R}$; the electromagnetic two-form and its potential and index dictionary; the particle functional $S_{\mathrm{p}\mathrm{t}}$ with its fixed-endpoint $C^2$ variations; and the adopted particle-sector stationary-action principle.

The particle functional is $S_{\mathrm{p}\mathrm{t}}[X]=-mc{\int }_a^b\sqrt{-\eta (\dot{X},\dot{X})}ds+q{\int }_a^b{A}_{\mu }(X){\dot{X}}^{\mu }ds$ on the declared class, with admissible variations the fixed-endpoint $C^2$ variations $h(a)=h(b)=0$ of Fixed-endpoint variation of the relativistic particle functional (Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional (c)).

The curve-variation lemma states that for a $C^2$ future timelike curve $X:[a,b]\to A$ and a $C^2$ covector field $A_{\mu }$ on a neighbourhood of its image, for every fixed-endpoint $C^2$ variation the curve $X+\epsilon h$ remains future timelike for all sufficiently small $\mathrm{\mid }\epsilon \mathrm{\mid }$, and stationarity of the displayed functional under all such variations is equivalent to $m\frac{d{U}_{\nu }}{d\tau }=q{F}_{\nu \mu }{U}^{\mu }$, $\frac{d\tau }{ds}=\frac{\sqrt{-\eta (\dot{X},\dot{X})}}{c}$, $F=dA$, with the four-velocity $U$ and proper time $\tau$ of the observer-decomposition theorem (Fixed-endpoint variation of the relativistic particle functional).

A massive worldline is a future timelike worldline of class $C^k$, $k\ge 1$; its four-velocity $U$ and four-acceleration $A$ satisfy $\eta (U,U)=-c^2$ and $\eta (U,A)=0$ by the observer-decomposition theorem (Worldlines, four-velocity, chart readings and the frame/observer distinction (a)-(b)).

The observer-decomposition theorem states (clause (a)) that for a $C^1$ future timelike worldline with four-velocity $U$ one has $\eta (U(s),U(s))=-c^2$ for every $s$ (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition).

The antisymmetric-contraction lemma states that for a skew form $F$ on the four-dimensional Minkowski vector space there is exactly one linear $\mathrm{\Phi }$ with $\eta (\mathrm{\Phi }(u),w)=F(w,u)$, and that in the fixed basis $\mathrm{\Phi }(u{)}^{\mu }={\sum }_{\alpha ,\beta }{\eta }^{\mu \alpha }{F}_{\alpha \beta }{u}^{\beta }$ (Antisymmetric contractions vanish and $m{U}^{\mathrm{\prime }}=q\mathrm{\Phi }(U)$ preserves $\eta (U,U)$).

The two-form definition states that the raised-index operator $\mathrm{\Phi }$ of the antisymmetric-contraction lemma satisfies $\mathrm{\Phi }(u{)}^{\mu }={\eta }^{\mu \alpha }{F}_{\alpha \beta }{u}^{\beta }$, and that if a $C^1$ covector field satisfies $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$ then the recovered fields obey $E=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A$ and $B=\mathrm{curl}A$ with $\phi :=-c{A}_0$ (The electromagnetic two-form, its observer decomposition and the four-current (c)-(d)).

The framework takes a real charge parameter $q\in \mathbb{R}$ in coulombs and a rest-mass parameter $m$ carrying the kilogram (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current (a), Framework-local SI units, the vacuum constants and the exact/measured status of their values (c)).

The relativistic Lorentz-force theorem states that, for a future timelike $C^2$ massive worldline in a prescribed $C^1$ field generated externally to the particle, the adopted equation of motion $m\frac{dU}{d\tau }=q\mathrm{\Phi }(U)$, $\eta (\mathrm{\Phi }(u),w)=F(w,u)$ implies, wherever the worldline is $C^2$, the coordinate force and power laws $\frac{dp}{dt}=q(E+v\times B)$ and $\frac{d(\gamma m{c}^2)}{dt}=qE\cdot v$, with $p=m\gamma v$ and $U=\gamma (c,v)$ (The four-force equation reproduces the coordinate Lorentz force and power).

The adopted particle-sector principle states that the motion is dynamically admissible exactly when the worldline is stationary for the particle functional under all fixed-endpoint $C^2$ variations (Adopt stationary Maxwell action as alternative physical formulation (ii)).