Mathematical Sciences: Improved Long-Time Estimates for the Stability of Motion in Hamiltonian Models of Accelerator Dynamics

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The investigator will develop realistic stability estimates "of Nekhoroshev type" for nearly integrable Hamiltonian models of physical systems, beginning with models for the motion of charged particles in accelerator storage rings. He uses results from averaging theory for ordinary differential equations and from a theory for the motion of energetic charged particles based on Nekhoroshev-type estimates. By combining recent progress in long-time estimates with quasiconvex unperturbed part (due to Lochak and Poeschel), and in averaging theory for ordinary differential equations (Ellison, Saenz, and Dumas), he expects to develop improved estimates for specific model systems. This expectation is based on the way averaging methods are embedded in Lochak's new approach: his methods reduce the problem --- in what is probably an optimal way --- to a high-order, single-phase, periodic averaging problem of the type for which improved estimates were recently developed. The proposed estimates could prove to be valuable tools in the development and design of increasingly complex particle accelerators. Such estimates might augment or even partly supplant some of the costly experimental methods now used to design and test accelerator lattices (the collections of multipole magnetic elements used to stabilize particle beams orbiting in accelerator storage rings). In view of the enormous development costs of the next generation of accelerators (such as the Superconducting Super-Collider, now in the initial phases of construction in Waxahachie, Texas, even as its development continues), even a small contribution to the design theory could result in a sizable savings of time and money. Moreover, estimates of the same type might be useful in much the same way in the design and development of "toroidal confinement fusion machines" (hot fusion reactors).