Title:

Interaction of two charges in a uniform magnetic field

The thesis starts with a short introduction to smooth dynamical systems and Hamiltonian dynamical systems. The aim of the introductory chapter is to collect basic results and concepts used in the thesis to make it self–contained. The second chapter of the thesis deals with the interaction of two charges moving in R2 in a magnetic field B. This problem can be formulated as a Hamiltonian system with four degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotational symmetry we reduce this Hamiltonian system to one with two degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain an integrable system. Furthermore, when the interaction potential is of Coulomb type, we prove that, for suitable regime of parameters, there are invariant subsets on which this system contains a suspension of a subshift of finite type. This implies non–integrability for this system with a Coulomb type interaction. Explicit knowledge of the reconstruction map and a dynamical analysis of the reduced Hamiltonian systems are the tools we use in order to give a description for the various types of dynamical behaviours in this system: from periodic to quasiperiodic and chaotic orbits, from bounded to unbounded motion. In the third chapter of the thesis we study the interaction of two charges moving in R3 in a magnetic field B. This problem can also be formulated as a Hamiltonian system, but one with six degrees of freedom. We keep the assumption that the magnetic field is uniform and the interaction potential has rotational symmetry and reduce this Hamiltonian system to one with three degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain a system with two degrees of freedom. Furthermore, when the interaction potential is chosen to be Coulomb we prove the existence of an invariant submanifold where the system can be reduced by a further degree of freedom. The reductions simplify the analysis of some properties of this system: we use the reconstruction map to obtain a classification for the dynamics in terms of boundedness of the motion and the existence of collisions. Moreover, we study the scattering map associated with this system in the limit of widely separated trajectories. In this limit we prove that the norms of the gyroradii of the particles are conserved during an interaction and that the interaction of the two particles is responsible for a rotation of the guiding centres around a fixed centre in the case of two charges whose sum is not zero and a drift of the guiding centres in the case of two charges whose sum is zero.
