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

Transport in near-integrable, but partially chaotic, 112 degree-of-freedom Hamiltonian systems is blocked by invariant tori and is reduced at 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 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”, action-gradient–minimizing (AGMin), quadratic-flux-minimizing (QFMin) and 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.

► We generalize Hamiltonian dynamics by defining three types of approximate orbits.
► We provide a generalization of action-angle coordinates in non-integrable systems.
► An application is to the description of plasmas in three-dimensional magnetic fields.