The Grad-Shafranov equation which determines the ideal axisymmetric hydromagnetic plasma equilibrium is transformed to a general curvilinear orthogonal system by the conformal mapping method. For the physical problems of plasma equilibrium, transformations which take into account the symmetry conditions and the constraints of several plasma configurations are investigated. For a number of such transformations the linear Grad-Shafranov equation is solved using the method of separation of variables. Old solutions are recovered and new analytic solutions in spherical, prolate and oblate spheroidal co-ordinates are deduced from which compact toroidal equilibria are derived. A formula for the average beta value of the determined equilibria is derived and the interval in which it lies is determined.
Axisymmetric hybrid Vlasov equilibria with applications to tokamak plasmas