Hamiltonian in Guiding Center Theory: A Symplectic Structure Approach

Abstract

The guiding center approximation represents a very powerful tool for analyzing and modeling a charged particle motion in strong magnetic fields. This approximation is based on conservation of the adiabatic invariant, magnetic moment. Hamiltonian equations for the guiding centre motion are traditionally intoduced using a non-canonical symplectic structure. Such approach requires application of non-canonical Hamiltonian perturbation theory for calculations of the magnetic moment corrections. In this study we present an alternative approach with canonical Hamiltonian equations for guiding centre motion in time-dependent electromagnetic fields. We show that the derived Hamiltonian decouples three types of motion (gyrorotation, field-aligned motion, and across-field drifts), and each type is described by a pair of conjugate variables. This form of Hamiltonian and symplectic structure allows simple introduction of adiabatic invariants and can be useful for analysis of various plasma systems.

Electron Resonant Interaction With Coherent ULF Waves: Hamiltonian Approach

“…There is more than one approach to derive the gyro‐averaged Hamiltonian describing particle drifts, and the most developed one is the noncanonical guiding center theory (Brizard & Chan, 2022; Cary & Brizard, 2009). However, we prefer to develop a canonical guiding center theory (Gardner, 1959; Neishtadt & Artemyev, 2020) that may provide simpler tools for investigation of resonant wave‐particle interactions (e.g., Artemyev et al., 2022b; Degeling & Rankin, 2008). We use Euler potentials α , β defined as B 0 = ∇ α × ∇ β (Stern, 1970).…”

On the Role of Whistler‐Mode Waves in Electron Interaction With Dipolarizing Flux Bundles

“…However, incorporating effects of electron‐scale wave perturbations to the electron motion using such guiding‐averaged equations is not straightforward. Because our final goal is to describe effects of electron resonant scattering by whistlers on electron interaction with a dipolarizing flux bundle, we need to derive a canonical set of guiding center equations (Gardner, 1959; Neishtadt & Artemyev, 2020). In this canonical set of Hamiltonian equations μ is a Hamiltonian variable, which is considered conserved (invariant) in the absence of waves and its conservation is violated when wave‐particle interactions occur.…”

“…This model is based on the canonical theory of adiabatic invariants and differs slightly from the more widespread models using the noncanonical theory (see discussion in the review by Cary & Brizard, 2009). The basic elements of the canonical theory are reviewed in Appendix A and in Neishtadt and Artemyev (2020). In Section3 we describe the inclusion of wave-particle resonant effects into the test particle simulation.…”