Nonselfintersecting magnetic orbits on the plane. Proof of Principle of the Overthrowing of the Cycles.

Abstract

Beginning from 1981 one of the present authors (S.Novikov) published a series of papers, (some of them in collaboration with I.Schmelzer and I.Taimanov) dedicated to the development of the analog of Morse theory for the closed 1-forms -- multivalued functions and functionals -- on the finite - and infinite-dimensional manifolds ({\bf Morse-Novikov Theory}). The notion of ``Multivalued action'' was understood and ``Topological quantization of the coupling constant'' for them was formulated by Novikov in 1981 as a Corollary from the requirement, that the Feinmann Amplitude should be one-valued on the space of fields-maps. Very beautiful analog of this theory appeared also in the late 80-ies in the Symplectic Geometry and Topology, when the so-called Floer Homology Theory was discovered. A very first topological idea of this theory, formulated in early 80-ies, was the so-called ``Principle of the Overthrowing of the Cycles''. It led to the results which were not proved rigorously until now. Our goal is to prove some of them. We study the motion of a classical charged particle on the Euclidean plane in a magnetic field orthogonal to this field. The trajectories of this motion can be characterized as extremals of the ``Maupertui--Fermat'' functional. We show that for any smooth everyvhere positive double periodic magnetic field for any fixed energy there exist at least two different periodic convex extremals, such that the value of the Maupertui-Fermat functional is positive for them. If all such extremals are nondegenerate in the sense of Morse in the space of nonparameterized curves then for any energy there exist at least 4 periodic convex extremals with the Morse indices (1,2,2,3).