Hamiltonian Theory of Adiabatic Motion of Relativistic Charged Particles

TL;DR

This paper develops a comprehensive Hamiltonian framework for the adiabatic motion of relativistic charged particles in varying electromagnetic fields, unifying and extending previous theories for all three adiabatic invariants.

Contribution

It introduces a unified Lie-transform perturbation approach in extended phase space to derive relativistic guiding-center equations and adiabatic invariants, including first-order corrections.

Findings

Derivation of relativistic guiding-center equations of motion.

Explicit expressions for second and third adiabatic invariants.

First-order corrections to adiabatic invariants for relativistic particles.

Abstract

A general Hamiltonian theory for the adiabatic motion of relativistic charged particles confined by slowly-varying background electromagnetic fields is presented based on a unified Lie-transform perturbation analysis in extended phase space (which includes energy and time as independent coordinates) for all three adiabatic invariants. First, the guiding-center equations of motion for a relativistic particle are derived from the particle Lagrangian. Covariant aspects of the resulting relativistic guiding-center equations of motion are discussed and contrasted with previous works. Next, the second and third invariants for the bounce motion and drift motion, respectively, are obtained by successively removing the bounce phase and the drift phase from the guiding-center Lagrangian. First-order corrections to the second and third adiabatic invariants for a relativistic particle are derived. These results simplify and generalize previous works to all three adiabatic motions of relativistic magnetically-trapped particles.