GITA                     Library "JINRLIB"                       

                                                                     You are 
    Authors: V.Basios, N.A.Chekanov, B.L.Markovski,                  
             V.A.Rostovtsev, S.I.Vinitsky                            visitor here.
    Language: Reduce                                                 
    
       A REDUCE PROGRAM FOR THE NORMALIZATION OF POLYNOMIAL HAMILTONIANS

    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.