Stability of relative equilibria in arbitrary axisymmetric gravitational and magnetic fields

Relative equilibria occur in a wide variety of physical applications, inclu ding celestial mechanics, particle accelerators, plasma physics, and atomic physics. We derive sufficient conditions for Lyapunov stability of circula r orbits in arbitrary axisymmetric gravitational (electrostatic) and magnet ic fields, including the effects of local mass (charge) and current density . Particularly simple stability conditions are derived for source-free regi ons, where the gravitational field is harmonic (del(2)U = 0) or the magneti c field irrotational (del x B = 0). In either case the resulting stability conditions can be expressed geometrically (coordinate-free) in terms of dim ensionless stability indices. Stability bounds are calculated for several e xamples, including the problem of two fixed centers, the J(2) planetary mod el, galactic disks, and a toroidal quadrupole magnetic field.