appendix4.tex

\chapter{Velocity-space and Energy-pitch space}\label{app:ep2vs}
There are two choices of coordinates for representing the velocity components of the fast-ion phase-space: the velocities parallel and perpendicular to the plasma current, ``velocity-space'', and the energy and pitch of the fast ion, ``energy-pitch'' space.
The two coordinates are related in the following way,
$$ E = \frac{m}{2}\left ( v_{\parallel}^2 + v_{\perp}^2 \right) \qquad
p = \frac{v_{\parallel}}{\sqrt{v_{\parallel}^2 + v_{\perp}^2}}$$
where $m$ is the mass of the fast ion, $E$ and $p$ are the energy and pitch of the fast-ion and $v_\parallel$ and $v_\perp$ are the parallel and perpendicular components of the velocity with respect to the plasma current.
The inverse relations are given by
$$ v_\parallel = \sqrt{\frac{2E}{m}} p \qquad
v_\perp = \sqrt{\frac{2E}{m}(1-p^2)}$$
The Jacobian of the transformation from energy-pitch to velocity-space and vis versa is given by, 
$$J_{Ep\rightarrow VS} = \frac{m v_\perp}{\sqrt{v_{\parallel}^2 + v_{\perp}^2}} \qquad
J_{VS\rightarrow Ep} = {\frac{1}{m \sqrt{1-p^2}}}\,. $$