Lorentz equation

When radiation reaction is weak (the radiation power is much smaller than the energy gain power), the motion of charged particles is governed by the Newton-Lorentz equation:

$$\frac{d{\mathbf{p}}}{d t} = \frac{q}{m}(\mathbf{E} + \boldsymbol{\beta}\times\mathbf{B}),$$

$$\frac{d\mathbf{x}}{d t} = \frac{\mathbf{p}}{\gamma}$$

where $\mathbf{p} \equiv \gamma m \mathbf{v}$, $\mathbf{x}$, $q$, $m$, $\gamma$, $\mathbf{v}$, and $\boldsymbol{\beta} \equiv \mathbf{v}/c$ are the momentum, position, charge, mass, Lorentz factor, velocity, and normalized velocity of the particle, respectively. These coupled equations are discretized using a leapfrog algorithm as

$$\frac{\mathbf{p}^{n+1/2} - \mathbf{p}^{n-1/2}}{\Delta t} = \frac{q}{m}\left(\mathbf{E}^n + \frac{\mathbf{p}^n}{\gamma^n} \times \mathbf{B}^n\right),$$

$$\frac{\mathbf{x}^{n+1} - \mathbf{x}^n}{\Delta t} = \mathbf{v}^{n+1/2},$$

and solved using the standard Boris rotation:

$$\mathbf{p}^{n-1/2} = \mathbf{p}^- - \frac{q\Delta t}{2m}\mathbf{E}^n,$$

$$\mathbf{p}^{n+1/2} = \mathbf{p}^+ + \frac{q\Delta t}{2m}\mathbf{E}^n,$$

$$\mathbf{p}' = \mathbf{p}^- + \mathbf{p}^- \times \mathbf{t},$$

$$\mathbf{p}^+ = \mathbf{p}^- + \mathbf{p}' \times \mathbf{s},$$

$$\mathbf{t} = \frac{q\Delta t}{2m\gamma^n}\mathbf{B}^n,$$

$$\mathbf{s} = \frac{2\mathbf{t}}{1+t^2},$$

where $\gamma^n = \sqrt{1+\mathbf{p}^2}$. The update in momentum and position are asynchronized by half a time step, i.e., a leapfrog algorithm is used here. This leapfrog algorithm ensures the self-consistency of the momentum and position evolution.