= Instantaneous-boost effective action for a sine-Gordon kink
{title2=$L_{\rm eff}=-8\sqrt{1-\dot X^2}-2\pi Xf(t)$}
Substitute the instantaneous <Sine-Gordon kink> profile $\theta_K(\gamma(x-X))$ and prescribed velocity $-\gamma\dot X\theta_K'$ with $\gamma=(1-\dot X^2)^{-1/2}$ into the field density. For $A_1=0$, $A_0=xf(t)$, the integrals $\int(\theta_K')^2=8$ and $\int\theta_K'=2\pi$ give the displayed <collective-coordinate effective Lagrangian>. Its equation is $d(8\gamma\dot X)/dt=-2\pi f$. This is relativistic particle motion with mass $8$ and topological coupling charge $2\pi$. When $\gamma$ varies, the prescribed velocity omits a width-change contribution; <acceleration correction for an instantaneous boosted kink> describes the exact chain-rule distinction.
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