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Manuscript Title: A program to compute Birkhoff normal forms of symplectic maps in R**4.
Authors: A. Bazzani, M. Giovannozzi, E. Todesco
Program title: ARES
Catalogue identifier: ADAO_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 86(1995)199
Programming language: Fortran.
Computer: ALPHAVAX 3000/500.
Operating system: VMS, UNIX.
RAM: 1.2M words
Peripherals: disc.
Keywords: Particle physics, Elementary, Accelerator, Symplectic maps, Birkhoff normal forms, Resonances.
Classification: 11.10.

Nature of problem:
The computation of normal forms of symplectic maps provides informations on the structure of the phase space in a neighbourhood of a fixed point. This approach can be useful in beam dynamics to analyse the effects of nonlinearities of a magnetic lattice. The normal forms allow one to analytically optimize a magnetic lattice, constructing effective correction strategies for the multipolar errors.

Solution method:
The computation of normal forms is based on the solution of a conjugation equation for the initial map. It is convenient to represent both the normalizing transformation and the normal form by means of a Lie transformation. Then we can solve the equation by a perturbative method on the homogeneous orders of the Taylor expansion. The main algorithmic tool is the composition of two polynomial maps up to a given order in the Taylor expansion.

Running time:
17.7 sec. on ALPHAVAX 3000/500 or 19.5 sec. on HP 755 at perturbative order 10 in the quasi-resonant case.