LINA01                   Library "JINRLIB"                       

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    Authors: Yu.A.Ukolov, N.A.Chekanov, A.A.Gusev, V.A.Rostovtsev,   
             S.I.Vinitsky, Y.Uwano                                   visitor here.
    Language: Reduce vs.3.7.

       A REDUCE PROGRAM FOR THE NORMALIZATION OF POLYNOMIAL HAMILTONIANS

    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.