J. E. Howard

Generalizations of the Stormer problem for dust grain orbits

Abstract We investigate the generalized Stormer problem, which includes electromagnetic and gravitational forces on a charged dust grain near an axisymmetric planet. For typical charge-to-mass ratios neither force can be neglected. The effects of the different forces are discussed in detail. Thus, including the gravitational force gives rise to stable circular orbits lying in a plane entirely above/below the equatorial plane. A modified third Kepler’s law for these orbits is found and analyzed.

Nonkeplerian dust dynamics at Saturn

We calculate the spatial location of possible stable nonequatorial halo dust grain orbits about Saturn, with surface potential determined by local photoionization and magnetospheric charging currents. Stability loci are calculated for both dielectric and conducting grains, in prograde and retrograde orbits, with the sign of the charge determined by the plasma environment. The results show that very small (<100 nm) grains in positive retrograde orbits are most likely to be found by the Cassini orbiter, while negatively charged grains are dynamically excluded.

Discrete Virial Theorem

We reexamine the classical virial theorem for bounded orbits of arbitrary autonomous Hamiltonian systems possessing both regular and chaotic orbits. New and useful forms of the virial theorem are obtained for natural Hamiltonian flows of arbitrary dimension. A discrete virial theorem is derived for invariant circles and periodic orbits of natural symplectic maps. A weak and a strong form of the virial theorem are proven for both flows and maps. While the Birkhoff Ergodic Theorem guarantees the existence of the relevant time averages for both regular and chaotic orbits, the convergence is very rapid for the former and extremely slow for the latter. This circumstance leads to a simple and efficient measure of chaoticity. The results are applied to several problems of current physical interest, including the Henon— Heiles system, weak chaos in the standard map, and a 4D Froeschle map.

Linear stability of natural symplectic maps

Abstract We calculate linear stability boundaries for natural symplectic maps, which are symplectic mappings derived from Lagrangian generating functions having positive definite kinetic energy. Simplified stability conditions are obtained in terms of the Hessian of the potential and applied to a four-dimensional pair of coupled standard maps.

Stability of relative equilibria in arbitrary axisymmetric gravitational and magnetic fields

Relative equilibria occur in a wide variety of physical applications, including celestial mechanics, particle accelerators, plasma physics, and atomic physics. We derive sufficient conditions for Lyapunov stability of circular orbits in arbitrary axisymmetric gravitational (electrostatic) and magnetic fields, including the effects of local mass (charge) and current density. Particularly simple stability conditions are derived for source‐free regions, where the gravitational field is harmonic (∇2U = 0) or the magnetic field irrotational (∇ × B = 0). In either case the resulting stability conditions can be expressed geometrically (coordinate‐free) in terms of dimensionless stability indices. Stability bounds are calculated for several examples, including the problem of two fixed centers, the J2 planetary model, galactic disks, and a toroidal quadrupole magnetic field.

Halo orbits around saturn

We investigate the effects of planetary oblateness, quadrupole magnetic field, and solar radiation pressure on nonequatorial dust grain orbits about Saturn. Radiation pressure is found to be a strong perturber for small (≈ 100 nm) positively charged conducting grains

Unified Hamiltonian stability theory

We show that the spectral stability of Hamiltonian equilibria and periodic orbits may be analyzed by the same method. This is accomplished via the Cayley transform, λ=(σ+1)/(σ−1), which maps the unit circle onto the imaginary σ-axis. The advantages and disadvantages of the new method over previous techniques are elucidated.

A simple reconnecting map

Generalized standard maps of the cylinder for which the rotation number is a rational function (a combination of the Fermi and Chirikov rotation functions) are considered. These symplectic maps often have degenerate resonant zones, and we establish two types resonance bifurcations: “loops” and “vortex pairs”. Both the border of chaos and the existence of the chaotic web are discussed. Finally the transition to global chaos for a generalized map is considered.