Pinheiro, Diogo (2006) Interaction of two charges in a uniform magnetic field. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Library of Congress Subject Headings (LCSH):	Magnetic fields, Dynamics, Hamiltonian systems
Official Date:	July 2006
Dates:	Date Event July 2006 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	MacKay, R. S. (Robert Sinclair)
Sponsors:	Fundação para a Ciência e a Tecnologia (FCT) (9SFRH/BD/9239/2002)
Extent:	ix, 169 leaves
Language:	eng

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