Lagrangian and Hamiltonian constraints for guiding-center Hamiltonian theories

TL;DR

This paper develops a comprehensive guiding-center Hamiltonian theory using Lie-transform perturbation, ensuring consistency through new Jacobian and Lagrangian constraints, and introduces a novel polarization term with implications for magnetic confinement.

Contribution

It presents a new guiding-center Hamiltonian theory with second-order accuracy, incorporating previously unexplored Jacobian and Lagrangian constraints, and identifies a new polarization term.

Findings

Derived a consistent guiding-center Hamiltonian up to second order.

Identified a new first-order polarization term in the phase-space Lagrangian.

Showed the new polarization simplifies the expression of the toroidal canonical momentum.

Abstract

A consistent guiding-center Hamiltonian theory is derived by Lie-transform perturbation method, with terms up to second order in magnetic-field nonuniformity. Consistency is demonstrated by showing that the guiding-center transformation presented here satisfies separate Jacobian and Lagrangian constraints that have not been explored before. A new first-order term appearing in the guiding-center phase-space Lagrangian is identified through a calculation of the guiding-center polarization. It is shown that this new polarization term also yields a simpler expression of the guiding-center toroidal canonical momentum, which satisfies an exact conservation law in axisymmetric magnetic geometries. Lastly, an application of the guiding-center Lagrangian constraint on the guiding-center Hamiltonian yields a natural interpretation for its higher-order corrections.