Stationary charged action and Lorentz force

Let $I=[a,b]$ and $J=[\tilde{a},\tilde{b}]$ be compact nondegenerate real intervals, let $\varphi :J\to I$ be an admissible parameter change from [A1], and let $h:I\to \mathbb{R}$ be continuous. By [F5], $h$ is integrable. Define $H(x):={\int }_a^{x}h(s)ds$; [F6] gives $H^{\mathrm{\prime }}=h$ on $I$, with one-sided derivatives at its endpoints. The chain rule [F3] applied to $G:=H\circ \varphi$ gives $G^{\mathrm{\prime }}(\lambda )=h(\varphi (\lambda )){\varphi }^{\mathrm{\prime }}(\lambda )$ on $J$. This derivative is continuous by [F4] and therefore integrable by [F5]. Since $\varphi (\tilde{a})=a$ and $\varphi (\tilde{b})=b$, [F7] yields ${\int }_{\tilde{a}}^{\tilde{b}}h(\varphi (\lambda )){\varphi }^{\mathrm{\prime }}(\lambda )d\lambda =G(\tilde{b})-G(\tilde{a})=H(b)-H(a)={\int }_a^{b}h(s)ds$. Thus the one-dimensional integral substitution used below holds for each continuous action integrand.

For any local fixed-endpoint variation ${\gamma }_{ϵ}:[a,b]\to O$ from [A1], composition with $\varphi$ gives a $C^2$ fixed-endpoint variation ${\gamma }_{ϵ}\circ \varphi$ on $J$. In chart components, for ${\tilde{\gamma }}^i(ϵ,\lambda ):={\gamma }^i(ϵ,\varphi (\lambda ))$, the first and second derivatives in $\lambda$ are $({\mathrm{\partial }}_s{\gamma }^i)\circ \varphi {\varphi }^{\mathrm{\prime }}$ and $({\mathrm{\partial }}_{ss}{\gamma }^i)\circ \varphi ({\varphi }^{\mathrm{\prime }}{)}^2+({\mathrm{\partial }}_s{\gamma }^i)\circ \varphi {\varphi }^{\mathrm{\prime }\mathrm{\prime }}$; the derivatives in $ϵ$ and the mixed derivative are $({\mathrm{\partial }}_{ϵ}{\gamma }^i)\circ \varphi$, $({\mathrm{\partial }}_{ϵϵ}{\gamma }^i)\circ \varphi$ and $({\mathrm{\partial }}_{ϵs}{\gamma }^i)\circ \varphi {\varphi }^{\mathrm{\prime }}$. The chain and product rules [F3], together with continuity under composition and products [F4], show these derivatives are continuous, so the family remains $C^2$. Since ${\varphi }^{\mathrm{\prime }}>0$, the composed curves remain future timelike and have the same endpoint events, and the chart-supported variation field remains supported in that chart. Bilinearity of $g$ gives $\sqrt{-g(({\gamma }_{ϵ}\circ \varphi {)}^{\mathrm{\prime }},({\gamma }_{ϵ}\circ \varphi {)}^{\mathrm{\prime }})}=(\sqrt{-g({\gamma }_{ϵ}^{\mathrm{\prime }},{\gamma }_{ϵ}^{\mathrm{\prime }})})\circ \varphi {\varphi }^{\mathrm{\prime }}$, and linearity of the covector $A$ gives $A(({\gamma }_{ϵ}\circ \varphi {)}^{\mathrm{\prime }})=(A({\gamma }_{ϵ}^{\mathrm{\prime }}))\circ \varphi {\varphi }^{\mathrm{\prime }}$. Both scalar integrands are continuous by [F4]; applying step 1.1 to each proves $S[{\gamma }_{ϵ}\circ \varphi ]=S[{\gamma }_{ϵ}]$ for every $ϵ$. Composition with ${\varphi }^{-1}$ is the inverse map on variation families. Thus the action functions and their first variations agree, so stationarity is invariant under every admissible reparameterization.

The segment in the statement is parameterized by proper time, and [A1] admits $\lambda =\tau$. If the same segment and variations are expressed in any other admissible parameter, step 2.1 proves that both the action and fixed-endpoint stationarity are unchanged. Therefore it suffices to test stationarity with $\lambda =\tau$. Under [A2], the curve, potential, regularity and endpoint hypotheses now match the local first-variation result [F2].

Applying [F2] under [A2], stationarity for every local fixed-endpoint variation is equivalent to $m({\mathrm{\nabla }}_{U}U{)}_a=q{F}_{ab}{U}^b$. Raising the covector index with the nondegenerate metric and using the definition of $F^{\mathrm{♯}}$ in [F1] gives the equivalent vector equation $m{\mathrm{\nabla }}_{U}U=q{F}^{\mathrm{♯}}(U)$.