A Variational Framework for Guiding-Center Kinetics, Anisotropic Equilibria, and Quasisymmetry in Stellarators

We present a variational framework in which (i) guiding-center kinetic theory, (ii) macroscopic force balance with gyrotropic/anisotropic pressure, and (iii) quasisymmetry (QS) constraints appear as different facets of a single structure.

Starting from a guiding-center Vlasov--Maxwell action, constrained variations yield the guiding-center kinetic equation and Maxwell equations.

Without phenomenological closure, momentum conservation yields macroscopic force balance $\mathbf{J} \times \mathbf{B}/c = \nabla\cdot\boldsymbol{\Pi}$, where $\mathbf{J}$ is the current density, $\mathbf{B}$ is the magnetic field, and $\boldsymbol{\Pi}$ is the gyrotropic stress tensor.

We connect QS to an integrability condition expressed in coordinate-free form via $$f_T \equiv \nabla\psi \cdot \bigl(\nabla B \times \nabla(\mathbf{B} \cdot \nabla B)\bigr) = 0,$$ where $\psi$ is the flux surface label, and show how this condition leads to solvability constraints on anisotropy closely related to those found in a recent constrained Kruskal--Kulsrud variational formulation.