How to yield the relativistic equations of motion for a particle in a plasma non-dimensionless The relativistic equations equations of motion for the ions are:
$$
\frac{d\vec{p}}{dt} \ = \ q (\vec{E}+ \vec{u} \times \vec{B}), \ \vec{p} \ = \ m \gamma\vec{v}
$$
In many papers discussing test particle simulation in MHD the authors write the equations of motion in a non-dimensionless form. What is the exact process to yield the above dimensional relativistic equations of motion, dimensionless?
Using quantities such as

*

*r$_{0}$ (length): skin depth, $v_{A}/ \Omega_{c} = c/ \Omega_{p}$

*m$_{0}$ (mass): ion mass, m$_{i}$,
m$_{0}$ = m$_{p}$

*t$_{0}$ (time): inverse gyrofrequency, $1/ \Omega_{c}$, $t_{0} = 1/\Omega_{c}$

*v$_{0}$ (velocity): alfven speed, $v_{A}$, $v_{0} = v_{A}$

*B$_{0}$ (magnetic field): $B_{0}$

*E$_{0}$ (electric field): $v_{A}*B_{0}/c$

*E$_{0}$ = $v_{A}*B_{0}/c$
where, $v_{A}$ is the alfven speed, $\Omega_{c}$ the cyclotron frequency, $\Omega_{p}$ plasma frequency, $B_{0}$ the background magnetic field
 A: I'll take you through the steps I would take for this particular equation, and then try and explain my thought process at the end.
Let's define a normalised velocity, $\bar{v} = v / v_0$, and normalised mass, $\bar{m} = m / m_0$. From this we can define a normalised momentum, $\bar{p} = p / p_0$, where $p_0 = m_0 v_0$. To make the left hand side of the equation dimensionless, we also need to define a reference timescale such that $\bar{t} = t / t_0 = t \omega$. We can now rewrite the left hand side in terms of dimensionless and reference quantities:
$$
\frac{dp}{dt} = \frac{d \bar{t}}{dt}\frac{d (p_0\bar{p})}{d\bar{t}} = \omega p_0 \frac{d\bar{p}}{d\bar{t}}
$$
We can now rewrite the original equation:
$$
\frac{d\bar{p}}{d\bar{t}} = \frac{1}{\omega p_0}q\left(\vec{E} + \vec{v} \times \vec{B}\right)
$$
Note the right hand side now has units of Coulombs, so let's define a dimensionless charge, $\bar{q} = q / q_0$, to give:
$$
\frac{d\bar{p}}{d\bar{t}} = \frac{q_0}{\omega p_0}\bar{q}\left(\vec{E} + \vec{v} \times \vec{B}\right) = \bar{q} \left(\frac{q_0}{\omega p_0}\vec{E} + \frac{q_0}{\omega m_0} \bar{v} \times \vec{B}\right)
$$
The quantities $q_0 / \omega p_0$ and $q_0 / \omega m_0$ have inverse dimensions of electric and magnetic field respectively, so we have a dimensionless equation:
$$
\frac{d\bar{p}}{d\bar{t}} = \bar{q}\left(\bar{E} + \bar{v} \times \bar{B}\right)
$$
where:
$$
\bar{E} = \frac{E}{E_0} = \frac{q_0 E}{\omega p_0}
$$
and
$$
\bar{B} = \frac{B}{B_0} = \frac{B q_0}{\omega m_0}
$$.
So we've defined 4 reference quantities ($v_0, m_0, t_0, q_0$) and used them to define 3 further reference quantities ($E_0, B_0, p_0$). Note that this isn't unique. I could have instead used a reference length $L_0$, and calculate $t_0$ in terms of $L_0$ and $v_0$. The important thing is to not over specify the system - it wouldn't make sense to independently specify $t_0$, $L_0$ and $v_0$, for example.
You can tell from the presence of 4 SI base units (time, length, mass and current) you'll need to specify 4 reference quantities. Others will follow. However as I've shown here you don't have to specify reference quantities in these base units (I used velocity and charge rather than length and current).
