Language: Reduce

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Nature of problem:

The preparation of the normal Birkhoff-Gustavson form is the universal and consistent method of investigation of classical Hamiltonian systems [1-3].
Additionally in recent years the normal Birkhoff-Gustavson form has been used for the quantisation of classical trajectories with the aim of studying the quantum manifestation of the classical chaos in the quasi-classical approximation [4]. But the procedure of reducing the initial Hamiltonian to normal form [2] is very cumbersome, and one needs to perform it with the help of an algebraic manipulator.

Solution method:

The method of solution consists of performing a series of canonical transformations which reduce the given Hamiltonian to a normal form.
Then the formal integral of the motion is calculated up to terms of desired order by successively inverting those canonical transformations.

Restrictions:

The computer time grows rapidly with the number of degrees needed to solve to the required approximation, especially if one wants to calculate the formal integral of the motion. Time limits and available computer memory may cause restrictions.

Unusual features:

As is known, there are FORTRAN programs which implemented the Birkhoff-Gustavson algorithm [2,6]. Our GITA is the first REDUCE algebraic program which calculates the normal Birkhoff-Gustavson form and the integral of motion for the two-dimensional polynomial Hamiltonians.

Running time:

The running time varies from 6.3s to 405s for the fourth and the eighth degree of the approximations, respectively, for the well-known Henon-Heiles Hamiltonian [5].

See also: http://cpc.cs.qub.ac.uk/summaries/ADBW_v1_0.html

References:

1. G.D. Birkhoff, Dynamical systems (A.M.S. Colloquium Publications, New York, 1927).
2. F.G. Gustavson, Astron. J. v.71(1966)p.670.
3. V.I. Arnold, Mathematical methods of classical mechanics (Atomizdat, Moscow, 1974) (in Russian).
4. L.E. Reichl, The Transition to Chaos. In Conservative Classical Systems:
   Quantum Manifestations. (Springer-Verlag, New York, Inc. 1992).
5. M. Henon and C. Heiles, Astron. J. v.69(1964) p.73.
6. A. Giorgili, Comput. Phys. Commun. v.6(1979) p.331.
7. V.Yu. Gonchar, N.A. Chekanov, B.L. Markovski, V.A. Rostovtsev and S.I. Vinitsky,
   The program of analytical calculation of the normal Birkhoff-Gustavson form,
   in: Proc. 4th Intern. Conf. "Computer Algebra in Physical Research", eds. D.V. Shirkov, V.A. Rostovtsev and V.P. Gerdt (World Scientific, London, 1991) p.423.

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