Spinning tops and magnetic orbits

A number of directions were initiated by the author and his students in their papers of 1981–1982. However, one of them, concerning the properties of closed orbits on the sphere $S^2$ and in the groups $S^3$ and $\operatorname{SO}_3$, has not been sufficiently developed. This paper revives the discussion of these questions, states unsolved problems, and explains what was regarded as fallacies in old papers. In general, magnetic orbits have been poorly discussed in the literature on dynamical systems and theoretical mechanics, but Grinevich has pointed out that in theoretical physics one encounters similar situations in the theory related to particle accelerators such as proton cyclotrons. It is interesting to look at Chap. III of Landau and Lifshitz's Theoretical physics, vol. 2, Field theory (translated into English as The classical theory of fields [12]), where mathematical relatives of our situations occur, but the physics is completely different and there are actual strong magnetic fields.Bibliography: 12 titles.

spinning tops, magnetic orbits, self-intersections