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

Abstract

We present a variational framework in which (i) guiding-center kinetics, (ii) macroscopic force balance with gyrotropic pressure, and (iii) quasisymmetry (QS) appear as facets of a single structure. Starting from a guiding-center Vlasov--Maxwell action, constrained variations yields the guiding-center kinetic equation and Maxwell equations. Without phenomenological closure, momentum conservation yields $\mathbf{J}\times \mathbf{B}/c=\nabla \cdot \Pi$. This framework recovers the classical result that, for any equilibrium distribution built on the invariants of the reduced motion, the parallel projection of this balance holds identically, so that pressure anisotropy is a kinematic consequence of magnetic-moment conservation together with the variation of $B$ along field lines, rather than a modeling choice. We connect QS to an integrability condition expressed in coordinate-free form. We then derive an explicit expression for the current crossing the flux surfaces. Its flux-surface average vanishes identically in a QS field, but it vanishes pointwise only if $p_{\parallel }+p_{\perp }$ is a flux function. Combined with the parallel identity this forces $p_{\parallel }=p(\psi )+K(\psi )B^2$ and $p_{\perp }=p(\psi )-K(\psi )B^2$, consistent with the anisotropic equilibria obtained from a constrained Kruskal--Kulsrud functional, in which $(p_{\parallel }+p_{\perp })/2$ is a flux function [Phys. Rev. E 104, 015213 (2021)]. The structure arises from the kinetic moments without a Lagrange multiplier. For such equilibria the parallel current in a QS field follows in closed form, with a single resonance at $\iota =N/M$ in place of the dense set of resonant obstructions of a generic three-dimensional field. Finally, QS is a Noether symmetry, so its breaking is measured by the non-conservation of the associated momentum. We construct a distribution-weighted residual and relate it to existing QS metrics.