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 $\mathbf{J}\times \mathbf{B}\mathrm{/}c=\mathrm{\nabla }\cdot \mathbf{\Pi }$, where $\mathbf{J}$ is the current density, $\mathbf{B}$ is the magnetic field, and $\mathbf{\Pi }$ is the gyrotropic stress tensor. We connect QS to an integrability condition expressed in coordinate-free form via $$f_T\equiv \mathrm{\nabla }\psi \cdot (\mathrm{\nabla }B\times \mathrm{\nabla }(\mathbf{B}\cdot \mathrm{\nabla }B))=0,$$ 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.