R.S.MacKay
Charged particle motion in a strong magnetic field $B$ can be reduced in the adiabatic approximation to gyromotion with constant magnetic moment $\mu$ and a two degree of freedom system for a guiding centre. For low energy, the zeroth order approximation for the guiding centre is that it moves along fieldlines with parallel velocity $v$ conserving $H = \frac12 mv^2 + \mu |B|$. Typically, this consists of bouncing between two points on the fieldline with the same value of $|B|$ and smaller $|B|$ in between. But to next order the guiding centre drifts across the field and so transitions between different classes of bouncing motion typically occur.
We define a magnetic field to be isodrastic if no such transitions occur. An approximate analysis can be carried out using an adiabatic invariant $j$ for the bouncing motion, yielding isodrasticity if and only if homoclinic trajectories with the same energy have the same $j$ ($j$ is an action, hence the terminology (isodrastic = same action). Yet near the transitions, $j$ is not well conserved, so the analysis is inconsistent.
Here, we give an exact treatment of the problem, deriving a necessary and sufficient condition for isodrasticity in the form that for each type of transition an analogue of a Melnikov function be zero. To leading order in $\mu$, the Melnikov function gives the same result as the approximate analysis, but the exact version promises to be useful to solve the problem of confining the high-energy $\alpha$-particles produced by fusion of Deuterium and Tritium.
The talk will be illustrated by numerically computed examples.
Work with Nikos Kallinikos, Josh Burby, Shibabrat Naik and Elizabeth Paul.