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Symplectic Approximation of Hamiltonian Flows and Accurate Simulation of Fringe Field Effects

A Dissertation

In partial fulfillment of the requirements for the degree of Doctor of Philosophy from Michigan State University.

Abstract

In the field of accelerator physics, the motion of particles in the electromagnetic fields of periodic accelerators is usually approximated by the iteration of a symplectic map, which represents the system over short time, such as one turn around the accelerator. Unfortunately, due to the complexity of the systems, in practice only some approximation of the one-turn map can be computed, as, for example, the truncated Taylor series. To this end, simulation of the nonlinear dynamics consists, in general, of the following three steps:
  1. Computation of the truncated Taylor approximation of the one-turn map,
  2. Symplectification of the Taylor map, and
  3. Iteration of the resulting exactly symplectic map.
This dissertation addresses all three components of the process, with the emphasis being on developing new methods that allow long-term tracking as accurately and efficiently as possible.

Specifically, the contributions to the first step concern the fringe field effects. The truncated Taylor map should include every relevant effect, so that it is an accurate representation of the system over one turn. While it is straightforward to compute the truncated maps over the regions where the fields are independent of the longitudinal variable, it is not so anymore at the ends of the magnets, the so-called fringe field regions. We study fringe fields generically, to show their importance, and develop a method that allows "exact" fringe field map computation of superconducting magnets, for which the coils and the iron parts are represented by current wires. The theory is illustrated by a detailed study of fringe field effects on the nonlinear dynamics of the Large Hadron Collider at collision energy.

Many contributions are established to the second step. It is well known that the theory of generating functions of canonical transformations provides a possible symplectification method. It is shown that, by transforming the dynamical problem into a problem in symplectic geometry, a general theory can be developed, which leads to a set of infinitely many new types of generating functions. It follows that it is possible to use this extended set to produce symplectic maps, and to reduce the whole set of generators to classes that give the same symplectified map. Moreover, the effects of factorization of the linear parts on the outcome of symplectification were studied. A variety of examples show the performance of various generator types, from which it can be concluded that it is not only important to symplectify, but also to symplectify ``the right way". The precise meaning of the last statement is the subject of the optimal symplectification theory, which can be formulated using methods of symplectic topology. In particular, Hofer's metric allows the formulation of the optimality condition in a very general setting, and the solution leads to a generating function type (EXPO) that, in general, gives optimal results. In the proof, an interesting one-to-one correspondence between fixed points of symplectic maps and critical points of generating functions is developed, and a generalized Hamilton-Jacobi equation is derived.

Finally, as contribution to the third step, it is pointed out that the numerical method used to solve the implicit equations arising in the iteration of the symplectic maps makes a difference in the final results, and, in general, a fixed point iteration is more robust than the widely used Newton method.

B. Erdelyi (2001)

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