Abstract In this work, the theory of second gradient electrodynamics, which is an important example of generalized electrodynamics, is proposed and investigated. Second gradient electrodynamics is a gradient field theory with up to second-order derivatives of the electromagnetic field strengths in the Lagrangian density. Second gradient electrodynamics possesses a weak nonlocality in space and time. In the framework of second gradient electrodynamics, the retarded Green functions, first-order derivatives of the retarded Green functions, retarded potentials, retarded electromagnetic field strengths, generalized Lienard–Wiechert potentials and the corresponding electromagnetic field strengths are derived for three, two and one spatial dimensions. The behaviour of the electromagnetic fields is investigated on the light cone. In particular, the retarded Green functions and their first-order derivatives show oscillations around the classical solutions inside the forward light cone and it is shown that they are singularity-free and regular on the light cone in three, two and one spatial dimensions. In second gradient electrodynamics, the self-force and the energy release rate are calculated and the equation of motion of a charged point particle, which is an integro-differential equation where the infamous third-order time-derivative of the position does not appear, is determined.

中文翻译：

---

第二梯度电动力学：格林函数、波传播、正则化和自力

摘要 在这项工作中，提出并研究了二次梯度电动力学理论，它是广义电动力学的一个重要例子。二阶梯度电动力学是一种梯度场理论，具有拉格朗日密度中电磁场强度的二阶导数。第二梯度电动力学在空间和时间上具有弱的非定域性。在二次梯度电动力学的框架内，推导了延迟格林函数、延迟格林函数的一阶导数、延迟电位、延迟电磁场强度、广义 Lienard-Wiechert 电位和相应的电磁场强度，分别为三、二和一空间维度。在光锥上研究电磁场的行为。特别是，延迟格林函数及其一阶导数在前向光锥内的经典解周围显示振荡，并且表明它们在光锥上在三个、两个和一个空间维度上是无奇点和规则的。在二次梯度电动力学中，计算自力和能量释放率，并计算带电点粒子的运动方程，这是一个积分微分方程，其中没有出现臭名昭著的位置的三阶时间导数，决心，决意，决定。