Closed-Form Solution of Adiabatic Particle Trajectories in Axis-Symmetric Magnetic Fields

The dynamics of a low-energy charged particle in an axis-symmetric magnetic field is known to be a regular superposition of periodic-although possibly incommensurate-motions. The projection of the particle orbit along the two non-ignorable coordinates (x,y) may be expressed in terms of each other: y=y(x), yet-to our knowledge-such a functional relation has never been directly produced in literature, but only by way of a detour: first, equations of motion are solved, yielding x=x(t),y=y(t), and then one of the two relations is inverted, x(t)-> t(x). In this paper, we present a closed-form functional relation which allows us to express coordinates of the particle's orbit without the need to pass through the hourly law of motion.