# Variational principle for the parallel-symplectic representation of electromagnetic gyrokinetic theory

## TL;DR

This paper derives the full electromagnetic gyrokinetic Vlasov-Maxwell equations using a variational principle in the parallel-symplectic representation, ensuring exact energy conservation and including second-order effects.

## Contribution

It introduces a variational derivation of the nonlinear electromagnetic gyrokinetic equations with energy conservation and second-order current contributions.

## Key findings

- Exact energy conservation law derived from variational principle.
- Inclusion of second-order gyrocenter current density effects.
- Formulation accounts for perturbed magnetic field contributions.

## Abstract

The nonlinear (full-$f$) electromagnetic gyrokinetic Vlasov-Maxwell equations are derived in the parallel-symplectic representation from an Eulerian gyrokinetic variational principle. The gyrokinetic Vlasov-Maxwell equations are shown to possess an exact energy conservation law, which is derived by Noether method from the gyrokinetic variational principle. Here, the gyrocenter Poisson bracket and the gyrocenter Jacobian contain contributions from the perturbed magnetic field. In the full-$f$ formulation of the gyrokinetic Vlasov-Maxwell theory presented here, the gyrocenter parallel-Amp\`{e}re equation contains a second-order contribution to the gyrocenter current density that is derived from the second-order gyrocenter ponderomotive Hamiltonian.