For the coupled system of classical Maxwell-Lorentz equations we show that the quantities \begin{equation*}
\mathfrak{F}(\hat x, t)=\lim_{|x|\to \infty} |x|^2 F(x,t), \quad \mathcal{F}(\hat k, t)=\lim_{|k|\to 0} |k| \widehat{F}(k,t), \end{equation*} where $F$ is the Faraday tensor, $\hat{F}$ its Fourier transform in space and $\hat{x}:=\frac{x}{|x|}$, are independent of $t$. We combine this observation with the scattering theory for the Maxwell-Lorentz system due to Komech and Spohn, which gives the asymptotic decoupling of $F$ into the scattered radiation $F_{\mathrm{sc},\pm}$ and the soliton field $F_{v_{\pm\infty}}$ depending on the asymptotic velocity $v_{\pm\infty}$ of the electron at large positive (+), resp. negative (-) times. This gives a soft-photon theorem of the form \begin{equation*} \mathcal{F}_{\text{sc},+}(\hat{k}) - \mathcal{F}_{\text{sc},-}(\hat{k})= -( \mathcal{F}_{v_{+\infty}}(\hat{k})-\mathcal{F}_{v_{-\infty}}(\hat{k})), \end{equation*} and analogously for $\mathfrak{F}$, which links the low-frequency part of the scattered radiation to the change of the electron's velocity. Implications for the infrared problem in QED are discussed in the Conclusions.
Effective dynamics for solitons in the nonlinear Klein Gordon Maxwell system and the Lorentz force law
考虑四维Minkowski时空中的非线性Klein Gordon Maxwell系统。对于适当的非线性，系统存在孤子解，这些孤子解是T. D.李和合作者的纯复标量场。在小e极限下，我们发展了一个严格的孤子动力学微扰理论，其中e是电磁耦合常数。主要定理证明了孤子在背景电流产生的外电磁场扰动下的长时间稳定性，并计算了孤子的有效动力学为O（e）。有效动力学方程是相对论性粒子在经典电动力学中常见的洛伦兹力定律作用下的运动方程。
Long time behaviour of the solution of Maxwell's equations in dissipative generalized Lorentz materials (I) A frequency dependent Lyapunov function approach
Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel（艾克斯-马赛大学、法国国家科学研究中心、马赛中央理工学院、费斯尼尔研究所）；ENSTA Paris (Ecole Nationale Supérieure de Techniques Avancées)（巴黎国立高等先进技术学院）；CNRS (Centre National de la Recherche Scientifique)（法国国家科学研究中心）；INRIA (Institut National de Recherche en Informatique et en Automatique)（法国国家信息与自动化研究所）
