Nearly axisymmetric systems occur in many physical problems, including the motion of charged dust grains in planetary magnetospheres, plasma fusion devices, ion motion in Paul traps, and the complex orbits of asteroidal moons. In all these examples the motion is well described by an autonomous Hamiltonian with an approximately conserved canonical momentum, for which an averaged effective potential exists. For small deviations from axisymmetry the motion remains essentially two-dimensional, simply averaging over nonaxisymmetric perturbations. This effect has major consequences for the long time confinement of orbits, such as pairs of ions in a Paul trap. In this research we will develop new analytic methods for describing this transition from two- to three-dimensional behavior using canonical perturbation theory, Morse theory, and Birkhoff normal forms. Global bifurcations of critical points and Arnold diffusion will also be investigated. The results are expected to have immediate impact on several fields under active investigation, such as high-performance computers utilizing quantum computation and the confinement of charged dust grains under the influence of solar radiation pressure.
Axisymmetric systems have circular symmetry about an axis; a communication satellite in geosynchronous orbit about Earth is an example. In reality the Earth has small longitudinal deviations from sphericity, causing a geosynchronous satellite to wander in the sky rather than remain directly overhead, unless it is carefully placed at a stable longitude. Nevertheless, its orbit remains very nearly circular. This simple example illustrates a nearly axisymmetric system with quasi-two-dimensional behavior. This project investigates the detailed dynamics of such nearly axisymmetric systems. The resulting increased quantitative understanding will greatly aid in design, numerical simulation, and operation of such systems. Other (much more complex) examples include two-ion motion in ion traps, which are often constructed to be slightly elliptical rather than circular, and the orbits of small moonlets that have been recently discovered about many asteroids.
