Radial confinement of deeply trapped particles in a non-symmetric magnetohydrodynamic equlibrium

Quasisymmetry and omnigeneity of an equilibrium magnetic field are two distinct properties proposed to ensure radial localization of collisionless trapped particles in any stellarator. These constraints are incompletely explored, but have stringent restrictions on a magnetic geometry. This work employs an analytic approach to understand the implications of the constraints. The particles move in an intrinsically three dimensional equilibrium whose representation is given by earlier work of Weitzner and its extension here. For deeply trapped particles a local equilibrium expansion around a minimum of the magnetic field strength along a magnetic line suffices. This analytical non-symmetric equilibrium solution, enables explicit representation of the constraints. The results show that it is far easier to satisfy the omnigeneity condition than the quasisymmetry requirement. Correspondingly, there exists a large class of equilibrium close to quasisymmetry that remain omnigeneous while allowing inclusion of error fields, which may destroy quasisymmetry.