|Location:||MSRI: Simons Auditorium|
I define an /almost-invariant torus /in the phase space of a Hamiltonian system as one whose image under time evolution remains close (in some sense) to its initial configuration. Similarly, in measure-preserving 2-dimensional maps (such as the return map to a Poincaré section in the phase-space of a 1½ d.o.f. system) an /almost-invariant circle /remains close to its image after one iteration.
The current driver of this research is the need to quantify (and minimize) the departure from integrability of magnetic fields in non-axisymmetric toroidal plasma containment devices used in fusion power research. A more general motivation is to seek action-angle-like coordinates for non-integrable systems that are close in some sense to a nearby integrable system, not known in advance. The study of a single almost-invariant torus is a subset of these more global problems.
I discuss norms for defining closeness and strategies for constructing almost-invariant circles and tori. I find that discussing the problem in terms of action gradients provides a unifying theme.