Language: Reduce vs.3.7.

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Nature of problem:

The transformation bringing a given Hamiltonian function into the normal form namely, the normalization) is one of the conventional methods for non-linear Hamiltonian systems [1-6]. Recently, beyond classical mechanics, the normal form method has been applied to quantization of chaotic Hamiltonian systems with the aim of finding quantum signature of chaos [7]. Besides those utilities, the normalization requires quite cumbersome algebraic calculations of polynomials, so that the computer algebraic approach is worth studying to promote further investigations around the normalization together with the ones around the inverse normalization.

Solution method:

The canonical transformation proceeding the normalization is expressed in terms of the Lie transformation power series, which is also referred to as the Hori-Deprit transformation. After (formal) power series expansion as above, the fundamental equation of the normalization is solved for the normal form together with the generating function of transformation recursively from degree-3 to the degree desired to be normalized. The generating function thus obtained is applied to the calculation of (formal) integrals of motion.

Restrictions:

The computation time raises in a combinatorial manner as the desired degree of normalization does. Especially, such a combinatorial growth of computation is more significant in the inverse normalization than in the direct one. The hardware (processor and memory, for example) available for the computation may restrict either the degree of normalization or the computation time.

See also: http://cpc.cs.qub.ac.uk/summaries/ADUV_v1_0.html

References:

1. A.J. Lichtenberg, M.A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, Berlin-Heidelberg-New York, 1983).
2. G.D. Birkhoff, Dynamical Systems (A.M.S. Colloquium Publications, New York, 1927).
3. F. Gustavson, Astron. J., 71 (1966) 670.
4. G.I. Hori, Astron. Soc. Japan. 18 (1966) 287.
5. A. Deprit, Cel. Mech. 1 (1969) 12.
6. A.A. Kamel, Cel. Mech. 3 (1970) 90.
7. L.E. Reichl, The Transition to Chaos. Coservative Classical Systems: Quantum Manifestations (Springer, New York, 1992).
8. A.P. Markeev, Theoreticheskaja mekhanika (Nauka, Moskva,1990).
9. M.K. Ali, J. Math. Phys. 26 (1985) 2565.
10. A. Giorgilli, Comp. Phys. Com. 6 (1979) 331.
11. T. Uzer, R.A. Marcus, J. Chem. Phys. 81 (1984) 5013.

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