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Manuscript Title: Program for Birkhoff-Gustavson normal form for N degrees of freedom - BIRKHOFF 1.2.
Authors: M. Kaluza
Program title: BIRKHOFF version 1.2
Catalogue identifier: ACLL_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 74(1993)441
Programming language: Reduce.
Computer: SUN SPARCstation IPC.
Operating system: SunOS Release 4.1.1, UNIX.
RAM: 5000K words
Word size: 32
Keywords: Computer algebra, Polynomial hamiltonian, Birkhoff-gustavson Normal form, Integrals of motion, Chaos.
Classification: 5.

Nature of problem:
In classical as well as in quantum mechanics one wishes to obtain the properties of the solutions of equations of motion without actually solving the equations of motion explicitly. The Birkhoff-Gustavson normal form approach is a means of finding the approximate constants of motion of a hamiltonian system with polynomial potential, if they exist.

Solution method:
By a series of canonical transformations of coordinates and momenta the Hamiltonian can be brought into normal form. In terms of transformed coordinates and momenta, the integrals of motion and the normal-form Hamiltonian have simple structures. By truncating the procedure at a certain order the approximations to a normal-ordered Hamiltonian as well as to the integrals of motion are obtained. The study of consecutive approximations can reveal information on regularity and chaoticity of given orbits as well as provide excellent appproximations to the orbits in a regular regime. The semiclassical quantization of the normal form has been investigated and in some cases a very good agreement with quantum results was found. Particularly promising is the algebraic quantization of the normal form. BIRKHOFF 1.2 implements the original Gustavson algorithm of the normalization on the Hamiltonian for any number (N) of degrees of freedom in the symbolic algebra language REDUCE. The algorithm can treat both the resonant as well as nonresonant cases. The program consists of subroutines performing normalization of the Hamiltonian and associated canonical transformations of coordinates. A sample of a main program is included in Appendix A and shows the typical use of the routines. In Appendix B a part of the resulting output is presented. The program has been applied on a generalized Henon-Heiles system with two degrees of freedom, and compared with the tested numerical results of Gustavson. An excellent agreement was found in the numerical coefficients up to maximal order. Our program can be used for any polynomial Hamiltonian with any number of degrees of freedom.

The complexity of a problem is restricted by available memory and CPU time. The memory requirement is proportional to the size of the result- ing normal form Hamiltonian, which in turn depends on the complexity of the original Hamiltonian. The time requirement of the calculation grows as a**M where M is the order to which the normal form of Hamiltonian is sought. However, the base a depends strongly on the complexity of the Hamiltonian and the mode of calculation: native floating-point, n-digit precision floating point, exact integer arithmetic, or symbolic arithmetic.

Running time:
In Table 1 the running times for calculation of normal form and the approximate integral of motion are given for the generalized Henon- Heiles problem with two parameters as well as for the original Henon- Heiles problem. The memory requirement never exceeded the default amount of 5 Mbyte in the cases presented.