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CHARGED PARTICLES IN MAGNETIC FIELDS

The   note [49] contains the solution of a problem posed by Arnold on the existence of periodic homotopically trivial trajectories on a 2--torus in the presence of magnetic fields. The note proves that a certain Poincare section map  of the energy 3--torus preserves the center of mass in an appropriate measure induced by the Liouvillemeasure on the energy torus - this is the main step in the solution. In order to apply the Conley-Zehnder theorem the variable density of the invariant  measure  has to be uniformized (a relatively minor step); I found the uniformizing transformation by using the heat flow on the torus. 

A related note [64] describes the destruction of periodic orbits of a charged particle in a potential on the torus when the magnetic field is turned on. It turns out that the non-contractible orbits on the torus which survive in small magnetic fields must lose their equidistribution property,  in a certain  precise sense. The note contains an estimate on the disparity of the gaps in terms of a certain curvature.  Geometrically, the note exploits the fact that an associated Poincare map is non--exact: while preserving the area, it shifts the center of mass. I used some simple geometry to make an analytic estimate on the disparity of gaps.