Periodic orbits of an electric charge in a magnetic dipole field

Abstract

Periodic orbits in the St6rmer problem are studied using the symmetry lines of the Poincar6 map introduced by De Vogelaere. Many known facts are explained by mean of these lines. The dynamics of four special symmetry lines when the StSrmer parameter ")'1 changes is presented, and we obtain a clear global view of the structure of the simple periodic orbits and their bifurcations, including the asymmetrical ones. New asymmetrical multiple periodic orbits are obtained.

From period to quasiperiod to chaos: A continuous spectrum of orbits of charged particles trapped in a dipole magnetic field

“…The quasi-periodic orbits discussed above will have a Poincaré map surrounding this fixed point. One should note that due to the presence of unstable periodic orbits 5 , between contours of quasi-periodic orbits at a given energy, there are fixed points of unstable periodic orbits 19 . However, these unstable periodic orbits has zero measure in the phase space and may not be realized in nature.…”

“…To better understand the Van Allen radiation belt discovered in the mid-20th century, there have been rea) Also at Phillips Academy Andover, 180 Main Street, Andover, MA, 01810, USA b) Electronic mail: liusm@pmo.ac.cn newed interests in solving the Störmer's problem. In particular, periodic orbits in the equatorial plane have been explored extensively 7 , and there are also a few branches of periodic orbits in the Meridian plane at relatively high energies with distinct orbital shapes [3][4][5] . At low energies, due to the presence of approximate adiabatic invariant, the problem can be further simplified 13,14 , and the results, the so-called guiding center approximation, have been applied to studies of pulsars, radiation belts, and dynamics of particles in magnetic confinement devices 12,15,16 .…”

“…It is well-known that there are quasi-periodic orbits around stable periodic orbits in Hamiltonian systems with 2 degrees of freedom 1 and these quasi-periodic orbits are stable as well 2 . Since periodic orbits appear to have a negligible measure in the phase space [3][4][5] , they are difficult to realize in nature. Quasi-periodic orbits, on the other hand, may occupy a finite volume in the 4 dimensional (4D) phase space and be readily detectable.…”

Caos en la dinámica del cuerpo rígido

“…En la presente tesis se estudia el sistema dinámico del problema del cuerpo rígido con un punto fijo en un campo de gravedad constante usando la técnica de las líneas de simetría introducida por De Vogelaere [32,33], siendo ampliamente desarrollada y explotada por diversos autores. Entre los principales se encuentran Piña et al Quienes la aplicaron en el tratamiento del mapeo estándar y en el problema de Störmer [7,8,16,30]. De hecho este trabajo es una extensión de lo que investigaron Chavoya-Piña [8] quienes estudiaron solo un caso particular del problema del cuerpo rígido con un punto fijo en un campo de gravedad constante, aquí se atacaran diversas situaciones físicas no abordadas del problema.…”

Órbitas periódicas en el problema de tres cuerpos

“…Piña y Jiménez-Lara han usado el método de líneas de simetría para estudiar el mapeo estándar [50], el problema de Störmer [32], así como el problema de tres cuerpos con una ley del potencial inversa al cuadrado [31].…”