The Motion of a Charged Particle on a Riemannian Surface under a Non-Zero Magnetic Field

Abstract

In this paper we study the motion of a charged particle on a Riemmanian surface under the influence of a positive magnetic field B. Using Moser's Twist Theorem and ideas from classical pertubation theory we find sufficient conditions to perpetually trap the motion of a particle with a sufficient large charge in a neighborhood of a level set of the magnetic field. The conditions on the level set of the magnetic field that guarantee the trapping are local and hold near all non-degenerate critical local minima or maxima of B. Using symplectic reduction we apply the results of our work to certain S 1 -invariant magnetic fields on R 3 .

Motion of Charged Particles in ABC Magnetic Fields

“…In order to illustrate the analytical results obtained in the previous sections, we restrict our study to the case C = 0, and hence we consider the reduced Hamiltonian system (7).…”

“…Assume that B = 1, C = 0, and L = 0. Then, the reduced Hamiltonian system (5)- (7) does not admit an additional first integral F : T 2 × R 2 → R which is a rational function in the variables sin x, cos x, sin z, cos z, p x , and p z .…”

“…Given 0 < B < 1, we assume that the parameter L satisfies L = O(ε), L < 2B, and L < 1 − B. Then, if ε is sufficiently small, the set Γ L consists of two smooth closed curves, and there are two points on each curve such that there exist quasi-periodic solutions ( 2-dimensional invariant tori) for the reduced Hamiltonian (7) in an ε-neighborhood of each point.…”

“…In what follows we shall denote by X H the vector field defined by (7). Let us compute the equilibrium points of this system, i.e., the critical points of X H .…”

The Hamiltonian dynamics of magnetic confinement in toroidal domains

“…Much work has been done on the analysis of the global behavior of solutions to Lorentz equations and related problems, e.g. [3,4,9,11]. Many of the previous results concerning the classical system are based on a perturbative approach and are proven by applications of Moser's twist theorem for perturbations of integrable systems.…”