Controlled weak-field first variation of the relativistic force law and the leading Thomson-model cross section

[Uniform smallness of the velocity and of $\gamma -1$] Let $K\subset I$ be a nonempty compact interval containing $0$ in its interior (any compact subinterval of $I$ containing $0$ will do); the expansion hypothesis gives $\mathrm{\mid }{v}_{\eta }(t)\mathrm{\mid }\le \mathrm{\mid }\eta \mathrm{\mid }\mathrm{\mid }{\dot{r}}_1(t)\mathrm{\mid }+C_K{\eta }^2\le \mathrm{\mid }\eta \mathrm{\mid }(\mathrm{\parallel }{\dot{r}}_1{\mathrm{\parallel }}_K+C_K)$ for $\mathrm{\mid }\eta \mathrm{\mid }<\min \{{\eta }_0,1\}$ and $t\in K$, where $\mathrm{\parallel }{\dot{r}}_1{\mathrm{\parallel }}_K$ is finite by continuity on the compact $K$. Choose ${\eta }_1\in (0,\min \{{\eta }_0,1\})$ with ${\eta }_1(\mathrm{\parallel }{\dot{r}}_1{\mathrm{\parallel }}_K+C_K)\le c\mathrm{/}2$; then for $0<\mathrm{\mid }\eta \mathrm{\mid }<{\eta }_1$ every $t\in K$ has $\mathrm{\mid }{v}_{\eta }(t)\mathrm{\mid }\mathrm{/}c\le 1\mathrm{/}2$, so the low-speed clause of [F1] with $\delta =1\mathrm{/}2$ gives $\mathrm{\mid }{\gamma }_{\eta }(t)-1\mathrm{\mid }\le \frac{\mathrm{\mid }{v}_{\eta }(t){\mathrm{\mid }}^2}{2c^2(1-1\mathrm{/}4{)}^{3\mathrm{/}2}}\le C_1{\eta }^2$ for a constant $C_1$ depending only on $K,{\eta }_1$ and the data, uniformly in $t\in K$.

[Uniform convergence of the scaled momentum and of its derivative] With ${\gamma }_{\eta }=(1-\mathrm{\mid }{v}_{\eta }{\mathrm{\mid }}^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}$ and $P_{\eta }:=m{\gamma }_{\eta }{v}_{\eta }$, write $\frac{1}{\eta }{P}_{\eta }=m{\gamma }_{\eta }\frac{v_{\eta }}{\eta }=m\frac{v_{\eta }}{\eta }+m({\gamma }_{\eta }-1)\frac{v_{\eta }}{\eta }$; by the expansion hypothesis $\frac{v_{\eta }}{\eta }\to {\dot{r}}_1$ uniformly on $K$ with error $C_K\mathrm{\mid }\eta \mathrm{\mid }$, and by step 1.1 $\mathrm{\mid }{\gamma }_{\eta }-1\mathrm{\mid }\le C_1{\eta }^2$ while $\mid \frac{v_{\eta }}{\eta }\mid \le \mathrm{\parallel }{\dot{r}}_1{\mathrm{\parallel }}_K+C_K\mathrm{\mid }\eta \mathrm{\mid }$ is uniformly bounded, so $\frac{P_{\eta }}{\eta }\to m{\dot{r}}_1$ uniformly on $K$ as $\eta \to 0$ with $\eta \mathrm{\ne }0$. Moreover, dividing the exact coordinate force law of [F1] by $\eta$ gives, for $t\in K$ and $\eta \mathrm{\ne }0$, $\frac{1}{\eta }{P}_{\eta }^{\mathrm{\prime }}(t)=q[E_{\mathrm{i}\mathrm{n}\mathrm{c}}(r_{\eta }(t),t)+v_{\eta }(t)\times B_{\mathrm{i}\mathrm{n}\mathrm{c}}(r_{\eta }(t),t)]$, and this right-hand side converges uniformly on $K$ to $q{E}_{\mathrm{i}\mathrm{n}\mathrm{c}}(x_0,t)$: the spatial Lipschitz bound of [F2] on a compact tube around $\{x_0\}\times K$ gives $\mathrm{\mid }{E}_{\mathrm{i}\mathrm{n}\mathrm{c}}(r_{\eta }(t),t)-E_{\mathrm{i}\mathrm{n}\mathrm{c}}(x_0,t)\mathrm{\mid }\le C\mathrm{\mid }{r}_{\eta }(t)-x_0\mathrm{\mid }\to 0$ uniformly by the expansion, while $\mathrm{\mid }{v}_{\eta }(t)\times B_{\mathrm{i}\mathrm{n}\mathrm{c}}(r_{\eta }(t),t)\mathrm{\mid }\le \mathrm{\mid }{v}_{\eta }(t)\mathrm{\mid }\sup \mathrm{\mid }{B}_{\mathrm{i}\mathrm{n}\mathrm{c}}\mathrm{\mid }\le C_2\mathrm{\mid }\eta \mathrm{\mid }$ by [F2] and the bilinearity and norm bound of [F5].

[First-order equation] Fix $\eta \mathrm{\ne }0$ with $\mathrm{\mid }\eta \mathrm{\mid }<{\eta }_1$ and fix a component index $i$. The scalar map $t\mapsto (P_{\eta }{)}_i(t)$ is $C^1$ on $K$ with continuous derivative $(P_{\eta }^{\mathrm{\prime }}{)}_i=q\eta [E_{\mathrm{i}\mathrm{n}\mathrm{c}}+v_{\eta }\times B_{\mathrm{i}\mathrm{n}\mathrm{c}}{]}_i$, so $(P_{\eta }^{\mathrm{\prime }}{)}_i$ is Riemann integrable on every closed subinterval of $K$ by [F5]; splitting at $0$ and applying the second fundamental theorem [F5] to the intervals $[0,t]$ for $t\ge 0$ and $[t,0]$ for $t\le 0$ (with the orientation convention of the displayed one-sided differences) gives, for every $t\in K$, $\frac{(P_{\eta }{)}_i(t)-(P_{\eta }{)}_i(0)}{\eta }={\int }_0^t\frac{(P_{\eta }^{\mathrm{\prime }}{)}_i(s)}{\eta }ds=q{\int }_0^t[E_{\mathrm{i}\mathrm{n}\mathrm{c}}(r_{\eta }(s),s)+v_{\eta }(s)\times B_{\mathrm{i}\mathrm{n}\mathrm{c}}(r_{\eta }(s),s){]}_{i}ds$. By the uniform convergence of step 2.1 applied componentwise, the integrands converge uniformly on $K$ to $q{E}_{\mathrm{i}\mathrm{n}\mathrm{c}}(x_0,\cdot {)}_i$, so the integral bound of [F5] lets $\eta \to 0$: the left side tends to $m[({\dot{r}}_1{)}_i(t)-({\dot{r}}_1{)}_i(0)]$ and the right side to $q{\int }_0^t{E}_{\mathrm{i}\mathrm{n}\mathrm{c}}(x_0,s{)}_{i}ds$. The fundamental theorem [F5] differentiates the right side in $t$ to $q{E}_{\mathrm{i}\mathrm{n}\mathrm{c}}(x_0,t{)}_i$, and the left side is differentiable with derivative $m({\ddot{r}}_1{)}_i(t)$; hence $m{\ddot{r}}_1(t)=q{E}_{\mathrm{i}\mathrm{n}\mathrm{c}}(x_0,t)$ componentwise on $K$. Every point $t\in I$ lies in such a compact $K\subset I$ containing $0$, so the identity holds on all of $I$ and is clause (a).

[Harmonic classification and the induced dipole] Equation (a) reads ${\ddot{r}}_1(t)=\mathrm{Re}(w{e}^{-i\omega t})$ on $I$ with $w=(q\mathrm{/}m)e{e}^{i\kappa \cdot x_0}$ by [F2]. By the local-clause of the harmonic lemma [F3], on $I$ one has $r_1=z_p+a+bt$ for a unique pair $(a,b)$, the same formula defines the unique global extension $Z$ of $r_1$, and $b=0$ exactly when $Z$ is $T$-periodic, equivalently when $r_1^{\mathrm{\prime }}(0)=z_p^{\mathrm{\prime }}(0)=\mathrm{Re}(iw\mathrm{/}\omega )$ (differentiating $z_p$ at $0$ with the rules of [F5]). Preparation (P) therefore says exactly $b=0$, so on $I$ we have $r_1=z_p+a$ with $a$ the mean of the periodic extension over every complete period by [F3]. Then $q(r_1-a)=q{z}_p(t)=\mathrm{Re}(-qw{\omega }^{-2}{e}^{-i\omega t})=\mathrm{Re}(p_0{e}^{-i\omega t})$ with $p_0=-qw\mathrm{/}{\omega }^2=-\frac{q^2}{m{\omega }^2}e{e}^{i\kappa \cdot x_0}$, because $w=(q\mathrm{/}m)e{e}^{i\kappa \cdot x_0}$; writing $e^{\mathrm{\prime }}=e{e}^{i\kappa \cdot x_0}$ one has $\mathrm{\mid }{e}^{\mathrm{\prime }}\mathrm{\mid }=\mathrm{\mid }e\mathrm{\mid }$ by [F5], so $p_0=\alpha e^{\mathrm{\prime }}$ with $\alpha =-\frac{q^2}{m{\omega }^2}$. The arbitrary mean $a$ is retained and no mean or periodicity of the nonlinear family is asserted; this is clause (b).

[Thomson cross section] Apply the cross-section formula of [F4] with the incident amplitude $e$ and dipole amplitude $p_0$ of step 4.1: $\sigma =\frac{{\omega }^4\mathrm{\mid }{p}_0{\mathrm{\mid }}^2}{6\pi {\epsilon }_0^2{c}^4\mathrm{\mid }e{\mathrm{\mid }}^2}$, and $\mathrm{\mid }{p}_0{\mathrm{\mid }}^2=\frac{q^4}{m^2{\omega }^4}\mathrm{\mid }e{\mathrm{\mid }}^2$ because $\mathrm{\mid }{e}^{i\kappa \cdot x_0}\mathrm{\mid }=1$; substituting cancels ${\omega }^4$ and $\mathrm{\mid }e{\mathrm{\mid }}^2$ and gives ${\sigma }_{\mathrm{T}}=\frac{q^4}{6\pi {\epsilon }_0^2{m}^2{c}^4}$. Multiplying out, $\frac{8\pi }{3}(\frac{q^2}{4\pi {\epsilon }_{0}m{c}^2}{)}^2=\frac{8\pi }{3}\cdot \frac{q^4}{16{\pi }^2{\epsilon }_0^2{m}^2{c}^4}=\frac{q^4}{6\pi {\epsilon }_0^2{m}^2{c}^4}$, the same value, and it depends neither on $\omega$ nor on the phase factor beyond the unimodular $e^{\mathrm{\prime }}$; this is clause (c).

[Scope and choice accounting] Clause (d) records the exact content: the result is a conditional first variation of the adopted external-force law under the stated family and preparation hypotheses, with the affine drift retained in general and set to zero only by (P) for (b) and (c); no existence, persistence, stability or self-consistency of the family, no self-force, radiation reaction or material content, and no empirical, atomic or frequency-response claim is made. The ${\eta }^2$ scaling statement is a remark on the leading model: for nonzero $\eta$ the incident amplitude $\eta e$ and the induced amplitude $\eta p_0$ scale together, so the ratio defining ${\sigma }_{\mathrm{T}}$ is unchanged; at $q=0$ the amplitude $p_0=-\frac{q^2}{m{\omega }^2}e{e}^{i\kappa \cdot x_0}$ and ${\sigma }_{\mathrm{T}}$ vanish by direct substitution without any division by $q$. The only choice principle used is the Countable Choice carried by the cross-section theorem [F4]; the uniform-limit, integral and harmonic-classification arguments use only the explicit suppliers [F1], [F3], [F5] and finite compact intervals.