Action-gradient-minimizing pseudo-orbits and almost-invariant tori

Transport in near-integrable, but partially chaotic, $1 1/2$ degree-of-freedom Hamiltonian systems is blocked by invariant tori and is reduced at \emph{almost}-invariant tori, both associated with the invariant tori of a neighboring integrable system. "Almost invariant" tori with rational rotation number can be defined using continuous families of periodic \emph{pseudo-orbits} to foliate the surfaces, while irrational-rotation-number tori can be defined by nesting with sequences of such rational tori. Three definitions of "pseudo-orbit," \emph{action-gradient--minimizing} (AGMin), \emph{quadratic-flux-minimizing} (QFMin) and \emph{ghost} orbits, based on variants of Hamilton's Principle, use different strategies to extremize the action as closely as possible. Equivalent Lagrangian (configuration-space action) and Hamiltonian (phase-space action) formulations, and a new approach to visualizing action-minimizing and minimax orbits based on AGMin pseudo-orbits, are presented.Comment: Accepted for publication in a special issue of Communications in Nonlinear Science and Numerical Simulation (CNSNS) entitled "The mathematical structure of fluids and plasmas : a volume dedicated to the 60th birthday of Phil Morrison

Parking a Spacecraft near an Asteroid Pair

This paper studies the dynamics of a spacecraft moving in the field of a binary asteroid. The asteroid pair is modeled as a rigid body and a sphere moving in a plane, while the spacecraft moves in space under the influence of the gravitational field of the asteroid pair, as well as that of the sun. This simple model captures the coupling between rotational and translational dynamics. By assuming that the binary dynamics is in a relative equilibrium, a restricted model for the spacecraft in orbit about them is constructed that also includes the direct effect of the sun on the spacecraft dynamics. The standard restricted three-body problem (RTBP) is used as a starting point for the analysis of the spacecraft motion. We investigate how the triangular points of the RTBP are modified through perturbations by taking into account two perturbations, namely, that one of the primaries is no longer a point mass but is an extended rigid body, and second, taking into account the effect of orbiting the sun. The stable zones near the modified triangular equilibrium points of the binary and a normal form of the Hamiltonian around them are used to compute stable periodic and quasi-periodic orbits for the spacecraft, which enable it to observe the asteroid pair while the binary orbits around the sun