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Exact Bounds on the Long Term Stability of Weakly Nonlinear Systems Applied to the Design of Large Storage Rings

Abstract

The motion in weakly nonlinear systems like circular accelerators and storage rings, like the Large Hadron Collider (LHC) at CERN and the terminated Superconducting Super Collider (SSC), can be described very well by high order Taylor maps representing the action on phase space. Recently it has been shown that such transfer maps can be obtained very elegantly and accurately using high order differential algebraic and automatic differentiation techniques in several variables. In typical examples, derivatives up to order ten in six variables are needed. This goes far beyond the abilities of other methods, which are limited to about order three.

Using nonlinear normal form transformations, the Taylor maps can be used to obtain families of six-dimensional approximate invariants of motion. In general, exact invariants can not be found, unless the motion is integrable, which in reality is rather unlikely. However, following arguments similar to those developed by Nekhoroshev, approximate invariants allow to find a direct lower bound on the time it takes particles to get lost, which is a crucial parameter for the design of large scale accelerators.

This method relies on bounding of a deviation function which describes fluctuations of the approximate invariants. Because the deviations from invariance are very irregular, a conventional estimate of the maximum of the six-dimensional deviation function is rather cumbersome and never completely accurate. On the other hand, interval methods allow an exact bound and in principle a very tight estimate of the maximum. To accomplish the task of maximizing the complicated functions involved, a very specialized interval maximizer had to be utilized. Because of the oscillatory structure of the deviation function and the large number of variables, this presents a nontrivial task which for the first time provides the missing link to a fully rigorous quantitative stability estimate.

All calculations are performed within the COSY language environment which provides an object-oriented structured language as well as an executer. The compiler and executer are written in FORTRAN 77 for easy portability. The ability to use dedicated data types for Taylor arithmetic and interval arithmetic is very helpful for the practical realization of the concepts discussed above.

After providing and overview over the problems of interest occurring in weakly nonlinear systems in general and specifically in particle accelerators, we will discuss the computation of approximate invariants of motion in detail. This is the basis for our computation of lower bounds on the number of turns particles survive in circular accelerators. The computer implementation and some results are discussed. Special emphasis will be put on the interval optimization procedure.

M. Berz, G. Hoffstätter, Interval Computations 2 (1994) 68-89

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