  We present a formalism for importing techniques from dynamical systems theory
in the study of three-dimensional magnetohydrodynamic (MHD) equilibria. By
treating toroidal angle as time, we reformulate the equilibrium equations as
hydrodynamic equations on the unit disc. They satisfy a variational principle
and comprise a Lie-Poisson Hamiltonian system. We use the variational principle
to find conservation laws for circulation, vorticity when pressure vanishes,
and energy in axisymmetric domains. Combining the Lie-Poisson structure with a
construction due to Scovel-Weinstein, we develop a theory of smoothed particle
magnetohydrostatics (SPMHS). SPMHS identifies a large class of exact
particle-swarm solutions of regularized spatial dynamics equations. The
regularization occurs at finer scales than the smallest scales within the
physical purview of ideal MHD. Crucially, the SPMHS equations of motion
comprise a finite-dimensional Hamiltonian system, which side steps perennial
roadblocks to a satisfactory theory of three-dimensional equilibria. In
large-aspect-ratio domains, we show the spatial dynamics equations comprise a
fast-slow system, where fast dynamics corresponds to the elliptic part of the
equilibrium equations and slow dynamics corresponds to the hyperbolic part.
Because the fast dynamics is formally normally hyperbolic, Fenichel theory
suggests the presence of an exact slow manifold for the spatial dynamics
equations that contains all physical solutions. We formally carry out reduction
to the slow manifold, obtaining a spatial dynamics formulation for equilibrium
Strauss reduced MHD at leading order and a hierarchy of corrections at any
desired order in perturbation theory. In large aspect-ratio domains, finding
periodic solutions of the slow manifold reduced equations represents a second
novel pathway to finding three-dimensional equilibria.
