Units

Note firstly that there are only 2 independent constants in the problem, $qB/m$ and $\gamma/m$, and that these constants have the units of inverse time; in fact the former is the cyclotron frequency and the latter is a damping rate. In general in any programming problem it pays to think carefully about the units to be used in the program. There are several reasons for this.

• If inappropriate units are used the program may not work at all. An example of this would be the use of SI units to study the dynamics of galaxies or to study atomic physics. In the former $R^3$ might easily arise and be bigger than the largest number representable on the machine, whereas in the latter $\hbar^4$ may be smaller than the smallest number on the machine and be set automatically to zero with disastrous consequences.
• The problem often has its own natural units and it makes sense to work in these units. This has the consequence that most of the numbers in your program will be of order unity rather than very large or very small.

In general you should look for the natural units of a problem and write your program appropriately. Note that these will generally not be SI or cgs. In the problem we are considering here there are 2 natural time scales, $m/qB$ and $m/\gamma$. If we decide to work in one of these, e.g. the cyclotron period $m/qB$, we can rewrite (1.43) in the simpler form

$${\d v_x\over \d t'} = +v_y - \left\vert\gamma\over qB\right\vert v_x$$ (1.44)

$${\d v_y\over \d t'} = -v_x - \left\vert\gamma\over qB\right\vert v_y$$ (1.45)

or perhaps

$${\d v_x\over \d t'} = -v_y - \left\vert\gamma\over qB\right\vert v_x$$ (1.46)

$${\d v_y\over \d t'} = +v_x - \left\vert\gamma\over qB\right\vert v_y$$ (1.47)

depending on the sign of $qB/m$. Here $t = t' \vert m/qB\vert$ and we have chosen our coordinate system such that the magnetic field, $\bi{B}$, is in the $z$-direction. Note, in addition, that choosing the units appropriately has eliminated all but one of the constants from the problem. This cuts down on superfluous arithmetic in the program.