The nonrelativistic Abraham–Lorentz equation as a restricted effective model

Work in one named laboratory chart with spatial coordinate space ${\mathbb{R}}^3$ and time coordinate $t\in \mathbb{R}$. Let $q\in \mathbb{R}\setminus \{0\}$ be the framework charge parameter, let $m>0$ be an externally supplied mass parameter, and take the SI constants $c>0$ and ${\epsilon }_0>0$ of the electromagnetic framework. Define the positive time constant

$\tau :=\frac{q^2}{6\pi {\epsilon }_{0}m{c}^3}>0.$

The units are $[q]=\mathrm{C}$, $[m]=\mathrm{k}\mathrm{g}$, $[c]=\mathrm{m}{\mathrm{s}}^{-1}$, and $[{\epsilon }_0]=\mathrm{F}{\mathrm{m}}^{-1}$, so $[\tau ]=\mathrm{s}$. A prescribed external force is a continuous map $f:\mathbb{R}\to {\mathbb{R}}^3$ with units $\mathrm{N}$. A particle trajectory in this model is a map $x:\mathbb{R}\to {\mathbb{R}}^3$ with three continuous time derivatives; its velocity and acceleration are $v(t):=\dot{x}(t)$ and $a(t):=\ddot{x}(t)$, with units $\mathrm{m}{\mathrm{s}}^{-1}$ and $\mathrm{m}{\mathrm{s}}^{-2}$. The term point particle means that the model assigns one position $x(t)$ and no spatial extent.

Adopt, as an additional effective model postulate, the equation

$m(a(t)-\tau \dot{a}(t))=f(t)(t\in \mathbb{R})\mathrm{.}$

Both $ma$ and $m\tau \dot{a}$ have units of force. The dot on $a$ means differentiation with respect to the laboratory time. This equation is not obtained by evaluating the particle's singular Maxwell self-field in the adopted Lorentz force law. The previously developed prescribed-source radiation model does not define that self-interaction; this postulate introduces a separate effective equation.

For physical use on a compact interval $I\subset \mathbb{R}$, assume $f$ is continuously differentiable there, choose dimensionless tolerances $\eta ,\delta \in (0,1)$ and a positive force scale $F_{\ast }$ with units $\mathrm{N}$, and state the chosen values. In this item, low-speed at tolerance $\eta$ means ${\sup }_{t\in I}\mathrm{\mid }v(t)\mathrm{\mid }\le \eta c$, and slowly varying relative to $\tau$ at tolerance $\delta$ means $\tau {\sup }_{t\in I}\mathrm{\mid }\dot{f}(t)\mathrm{\mid }\le \delta F_{\ast }$. These are explicit regime conventions for an application, not universal empirical thresholds. The ODE is an exact adopted equation on its mathematical domain; its physical interpretation is restricted to applications that state their regime and tolerances.

Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Charge parameter, frame charge and current densities, volume charge and flux, and the four-current

The Lorentz coupling: external force on a test charge and the continuum force and power densities

Self-force, runaways and finite-size regularizations are not supplied by this page