Research of James Meiss
|Much of the research listed below has received support from the National Science Foundation, most recently under grants DMS-0707659, DMS-1211350, CMMI-1447440, CMMI-1553297, and DMS-1812481 Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF. Support from the Simons Foundation, grant #601972, "Hidden Symmetries and Fusion Energy" is also gratefully acknowledged.|
|Though the flow of an incompressible vector field in 3 space, such as a magnetic field, is often thought of as a one-and-a-half
degree-of-freedom Hamiltonian system, i.e., H(q,p,t) with periodic time-dependence, we argue, in a paper with Josh Burby and Nathan Duignan, that it is more appropriate to think of it as a presymplectic system. There is a two-form that generates the dynamics, but it cannot be non-degenerate on a three-dimensional manifold.
We use this idea to reformulate the problem of magneto-hydro-static (MHS) equilibria, and to show that there exist normal form coordinates near a magnetic axis (a non-degenerate closed loop) that are analogous to Hamada and Boozer coordinates. The second invariant in this system corresponds to the current vector field (the diamagnetic current) and it is generated by the "Hamiltonian" given by the pressure.
Moving away from the integrable case, Nathan Duignan and I use this reformulation to compute asymptotic normal forms analogous to the Gustavson-Birkhoff form, near a magnetic axis both then the local rotational transform is irrational and rational. As shown in the figure at the right, this can be used in the near-resonant case to give a good approximation to the field lines, though the normal form necessarily breaks-down near the separatrix where it becomes chaotic.
|The concept of anti-integrability was introduced by Aubry and Abramovicci in 1983 for the standard map, viewed as a linear chain of particles connected by springs in a periodic potential. They reasoned that the integrable limit corresponded to vanishing potential energy, so that the springs dominated giving equal spacing at equilibrium. By contrast, anti-integrability corresponds to vanishing kinetic energy, so that particles sit at critical points of the potential. What is most interesting about this limit is that it is relatively easy, using a contraction mapping style argument, to show that AI states persist, and this gives conjugacy to a shift on a symbolic dynamics.
In the paper, Amanda Hampton and I generalize these ideas to the family of quadratic three-dimensional diffeomorphisms that were obtained in Lomeli & Meiss . We write the map as a third difference equation, and scale to isolate the nonlinear terms. A unique feature of this study is that the AI limit corresponds to a quadratic correspondence---a quadratic curve that corresponds to a one-dimensional dynamical system. We show that there are a number of parameter values for which a full shift on two-symbols exists at the AI limit and that these orbits can be continued away from the limit.
The figure at the right shows an orbit of a 3D quadratic map. continued away from the AI limit. At the limit, the orbit falls on the intersection of the two elliptic cylinders. As we move away from this limit, the orbit maintains some of this structure.