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

研究者将从加速器存储环中带电粒子运动的模型开始，为物理系统的近可积哈密顿模型开发“Nekhoroshev型”的现实稳定性估计。他使用了常微分方程的平均理论和基于nekhoroshev型估计的高能带电粒子运动理论的结果。通过结合准凸无扰动部分的长时间估计的最新进展（由于Lochak和Poeschel），以及常微分方程的平均理论（Ellison， Saenz和Dumas），他期望为特定模型系统开发改进的估计。这种期望是基于Lochak新方法中嵌入的平均方法的方式：他的方法可能以最优的方式将问题简化为高阶，单相，周期平均问题，这种类型的改进估计最近被开发出来。提出的估计可能被证明是开发和设计日益复杂的粒子加速器的有价值的工具。这样的估计可能会增加甚至部分取代目前用于设计和测试加速器晶格（用于稳定加速器存储环中轨道粒子束的多极磁性元件的集合）的一些昂贵的实验方法。考虑到下一代加速器的巨大开发成本（比如超导超级对撞机，目前正处于德克萨斯州瓦克哈奇的初始建设阶段，尽管它的开发仍在继续），即使对设计理论做出很小的贡献，也可能会节省大量的时间和金钱。此外，在设计和开发“环形约束核聚变机”（热核聚变反应堆）时，同样类型的估计也可能非常有用。

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).