Particle Motion

For a charged particle with mass \(m\) and charge \(q\) in the presence of electromagnetic fields, the equation of motion (EOM) is given by:

\[\begin{split}\frac{\mathrm{d} \mathbf{r}}{\mathrm{d} t} &= \mathbf{v},\\ \frac{\mathrm{d} \mathbf{v}}{\mathrm{d} t} &= \frac{q}{m} \left( \mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B} \right).\end{split}\]

Here we have defined the particle’s position \(\mathbf{r}\) and velocity \(\mathbf{v}\). The electric fields is \(\mathbf{E}\) and the magnetic fields is \(\mathbf{B}\) are measured at the particle’s position.

The equation of motion can be solved using a variety of numerical methods. The most common method is the Boris pusher, which is a second-order accurate method for solving the equation of motion.

Boris Pusher

Boris pusher is a second-order accurate symplectic integrator for solving the particle EOM. In order to use the Boris pusher, we define the particle velocity at half time steps \(\mathbf{v}^{n + 1/2}\) and particle position at full time steps \(\mathbf{r}^{n}\).

The velocity is updated using:

\[\begin{split}\mathbf{v}^{n + 1/2} &= \mathbf{v}^n + \frac{q}{2 m} \mathbf{E} \Delta t,\\ \mathbf{v}^{n + 1} &= \mathbf{v}^{n + 1/2} + \frac{q}{m} \left( \mathbf{v}^{n + 1/2} \times \mathbf{B} \right) \Delta t,\\ \mathbf{v}^{n + 3/2} &= \mathbf{v}^{n + 1} + \frac{q}{2 m} \mathbf{E} \Delta t.\end{split}\]

The position can then be updated trivially using:

\[\mathbf{r}^{n + 1} = \mathbf{r}^n + \mathbf{v}^{n + 1} \Delta t.\]