Some questions on the topology of the magnetic field

General requirements are considered for the topology of the field of a magnetic trap which is intended to realize a lengthy thermal insulation of plasma. The author studies the topologic structure of magnetic fields whose lines of force do not leave the boundaries of a finite volume. The necessary condition for the existence of such a field is the presence, among the integral surfaces of the field, of a smooth surfaoe homeomorphous to a torus, where the magnetic field strength does not vanish in any single point. Such fields can, in principle, be realized by means of a superconductor which bounds a toroidal cavity.The stability of the magnetic-field topology has been investigated with respect to small perturbations. The necessary and sufficient condition for the stability of the topology is that the rotation number change in the layer of toroidal magnetic surfaces. A method is proposed for establishing a trap with a magnetic-field strength increasing toward the periphery.Details of a mathematical character have been omitted since they are contained in the author's published papers of which the present paper is fundamentally a short summary.