Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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

Version 1 : Received: 3 September 2021 / Approved: 3 September 2021 / Online: 3 September 2021 (11:26:19 CEST)

How to cite: Sattin, F.; Escande, D. Closed-Form Solution of Adiabatic Particle Trajectories in Axis-Symmetric Magnetic Fields. Preprints 2021, 2021090061 (doi: 10.20944/preprints202109.0061.v1). Sattin, F.; Escande, D. Closed-Form Solution of Adiabatic Particle Trajectories in Axis-Symmetric Magnetic Fields. Preprints 2021, 2021090061 (doi: 10.20944/preprints202109.0061.v1).

Abstract

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 to express coordinates of the particle’ orbit without the need to pass through the hourly law of motion.

Keywords

Hamiltonian dynamics; Adiabatic invariants; Magnetized plasma; Particle trajectory; Canonical transformations

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