Hamiltonian approach to the existence of magnetic surfaces

Abstract

A method is devised to investigate the existence of magnetic surfaces and magnetohydrodynamic (MHD) plasma equilibria in 3-D toroidal geometry. The key feature of this method is the utilization of a Hamiltonian formulation of the lines of force. Expanding the contravariant components of the magnetic field and scalar pressure in distance ρ from the magnetic axis, the 1-D Hamiltonian for the lines of force is written out explicitly. The Hamiltonian is then transformed to action-angle variables. It is shown that the action J corresponds to pressure in the equilibrium problem. Specifically, it is shown that if J is an invariant, then constant pressure and hence magnetic surfaces exist. A procedure of repeated canonical transformations is formulated and carried out to displace the coordinate dependence in the Hamiltonian to successively higher order in the expansion parameter, and thus make J an increasingly better adiabatic invariant. Arising in each successive canonical transformation is a series of potentially resonant denominators, i.e., denominators that may vanish. These potential resonances are identified, their significance explicated, and methods of handling them suggested.