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The extended guiding-centre Lagrangian equations of motion are derived by the Lie-transform perturbation method under the assumption of time-dependent and inhomogeneous electric and magnetic fields that satisfy the standard guiding-centre space–time orderings. Polarization effects are introduced into the Lagrangian dynamics by the inclusion of the polarization drift velocity in the guiding-centre velocity and the appearance of finite-Larmor-radius corrections in the guiding-centre Hamiltonian and guiding-centre Poisson bracket. 

Extended guiding-center Vlasov–Maxwell equations are derived under the assumption of time-dependent and inhomogeneous electric and magnetic fields that obey the standard guiding-center space-timescale orderings. The guiding-center Vlasov–Maxwell equations are derived up to second order, which contains dipole and quadrupole contributions to the guiding-center polarization and magnetization that include finite-Larmor-radius corrections. Exact energy-momentum conservation laws are derived from the variational formulation of these higher-order guiding-center Vlasov–Maxwell equations. 

The problem of the charged-particle motion in crossed electric and magnetic fields is investigated, and the validity of the guiding-center representation is assessed in comparison with the exact particle dynamics. While the magnetic field is considered to be straight and uniform, the (perpendicular) radial electric field is nonuniform. The Hamiltonian guiding-center theory of charged-particle motion is presented for arbitrary radial electric fields, and explicit examples are provided for the case of a linear radial electric field. 

The problem of the charged-particle motion in an axisymmetric magnetic geometry is used to assess the validity of higher-order Hamiltonian guiding-center theory, which includes higher-order corrections associated with gyrogauge invariance as well as guiding-center polarization induced by magnetic-field non-uniformity. Two axisymmetric magnetic geometries are considered: a magnetic mirror geometry and a simple tokamak geometry. When a magnetically confined charged-particle orbit is regular (i.e., its guiding-center magnetic moment is adiabatically invariant), the guiding-center approximation, which conserves both energy and azimuthal canonical angular momentum, is shown to be faithful to the particle orbit when higher-order corrections are taken into account. 

Hamiltonian formulations of quasilinear theory are presented for the cases of uniform and nonuniform magnetized plasmas. First, the standard quasilinear theory of Kennel and Engelmann (Kennel, Phys. Fluids, 1966, 9, 2377) is reviewed and reinterpreted in terms of a general Hamiltonian formulation. Within this Hamiltonian representation, we present the transition from two-dimensional quasilinear diffusion in a spatially uniform magnetized background plasma to three-dimensional quasilinear diffusion in a spatially nonuniform magnetized background plasma based on our previous work (Brizard and Chan, Phys. Plasmas, 2001, 8, 4762–4771; Brizard and Chan, Phys. Plasmas, 2004, 11, 4220–4229). The resulting quasilinear theory for nonuniform magnetized plasmas yields a 3 × 3 diffusion tensor that naturally incorporates quasilinear radial diffusion as well as its synergistic connections to diffusion in two-dimensional invariant velocity space (e.g., energy and pitch angle). 

This Letter reports on a metriplectic formulation of a collisional, nonlinear full- f electromagnetic gyrokinetic theory compliant with energy conservation and monotonic entropy production. In an axisymmetric background magnetic field, the toroidal angular momentum is also conserved. Notably, a new collisional current, contributing to the gyrokinetic Maxwell–Ampère equation and the gyrokinetic charge conservation law, is discovered. 

The exact energy and angular momentum conservation laws are derived by the Noether method for the Hamiltonian and symplectic representations of the gauge-free electromagnetic gyrokinetic Vlasov–Maxwell equations. These gyrokinetic equations, which are solely expressed in terms of electromagnetic fields, describe the low-frequency turbulent fluctuations that perturb a time-independent toroidally-axisymmetric magnetized plasma. The explicit proofs presented here provide a complete picture of the transfer of energy and angular momentum between the gyrocentres and the perturbed electromagnetic fields, in which the crucial roles played by gyrocentre polarization and magnetization effects are highlighted. In addition to yielding an exact angular momentum conservation law, the gyrokinetic Noether equation yields an exact momentum transport equation, which might be useful in more general equilibrium magnetic geometries.