Abstract
A procedure for deriving homological equations of an arbitrary order is considered. Solutions of these equations are used in the iterative procedure of invariant normalization of the Hamiltonian in the neighborhood of equilibrium state. Specific features of the implementation of the algorithm for reducing to normal form using modern computer algebra systems are discussed. The normalization procedure is applied to the Hamiltonian of the Hill problem written in terms of scaled regular variables. The Hill problem normal form thus obtained can be used for finding analyticity sets of the normalizing transformation.
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Bruno, A.D., The Restricted 3-Body Problem: Plane Periodic Orbits, Moscow: Nauka, 1990; New York: de Gruyter, 1994.
Zhuravlev, V.F., Petrov, A.G., and Shunderyuk, M.M., Selected Problems of Hamiltonian Mechanics, Moscow: LENAND, 2015 [in Russian].
Bruno, A.D., Analytical form of differential equations, Tr. Mosk. Mat. O-va, 1972, vol. 26, pp. 199–236.
Birkhoff, G.D., Dynamical Systems, American mathematical society, New York, 1927.
Gustavson, F.G., On constructing formal integrals of a Hamiltonian system near all equilibrium point, Astron. J., 1966, vol. 71, no. 8, pp. 670–686.
Hori, G., Theory of General Perturbation with Unspecified Canonical Variable, Publ. Astron. Soc. Japan, 1966, vol. 18, no. 4, pp. 287–296.
Deprit, A., Canonical transformations depending on a small parameter, Celest. Mech., 1969, vol. 1, no. 1, pp. 12–30.
Haro, À., An algorithm to generate canonical transformations: application to normal forms, Physica D, 2002, vol. 167, pp. 197–217.
Petrov, A.G., Invariant normalization of nonautonomous Hamiltonian systems, J. Appl. Math. Mech., 2004, vol. 68, no. 3, pp. 357–367.
Zhuravlev, V.F., Invariant normalization of nonautonomous Hamiltonian systems, Prikl. Mat. Mekh., 2002, vol. 66, no. 3, pp. 356–365.
Kamel, A.A., Expansion formulae in canonical transformations depending on a small parameter Celest. Mech., 1969, vol. 1, no. 2, pp. 190–199.
Mersman, W., A new algorithm for the Lie transformation, Celest. Mech., 1970, vol. 3, no 1, pp. 81–89.
Giacaglia, G.E.O., Perturbation Methods in Non-Linear Systems, New York, Springer, 1972.
Nayfeh, A.H., Perturbation Methods, New York: Wiley, 1973.
Markeev, A.P., Libration Points in Celestial Mechanics and Astrodynamics, Nauka: Moscow, 1978 [in Russian].
Wolfram St., The Mathematica Book, Wolfram Media, Inc., 2003.
Prokopenya, A.N., Hamiltonian normalization in the restricted many-body problem by computer algebra methods, Program. Comput. Software, 2012, vol. 38, no. 3, pp. 156–169.
Shevchenko, I.I., Study of problems of stability and chaotic behavior in celestial mechanics, Doctoral (Math.) Dissertaion, St. Petersburg, Chief Astronomical Obsevatory, Russian Academy of Sciences, 2000, p. 257.
Burbanks, A.D., Wiggins, S., Waalkens, H., and Schubert, R., Background and Documentation of Software for Computing Hamiltonian Normal Forms, Bristol: School of mathematics, Univ. of Bristol, 2008.
Jorba, À., A methodology for the numerical computation of normal forms, centre manifolds and first integrals of Hamiltonian systems, Exp. Math., 1999, vol. 8, no. 2, pp. 155–195.
Bruno, A.D. and Batkhin, A.B., Algorithms and programs for calculating the roots of polynomial of one or two variables, Program. Comput. Software, 2021, vol. 47, no. 5, pp. 353–373.
Batkhin, A.B. and Batkhina, N.V., Hill’s Problem, Volgograd: Volgogradskoe nauchnoe Izdatel’stvo, 2009 [in Russian].
Szebehely, V., Theory of Orbits, the Restricted Problem of Three Bodies, New York: Academic, 1967.
Simó, C. and Stuchi, T.J., Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem, Physica D, 2000, vol. 140, pp. 1–32.
Batkhin, A.B., Web of families of periodic orbits of the generalized Hill problem, Dokl. Math. 2014, vol. 90, no. 2, pp. 538–544.
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The author is grateful to Professor A. D. Bruno for support and useful discussion of this paper.
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Translated by A. Klimontovich
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Batkhin, A.B. Symbolic Computation of a Homological Equation of an Arbitrary Order and Reduction of Hamiltonian System to Its Normal Form. Program Comput Soft 48, 65–72 (2022). https://doi.org/10.1134/S0361768822020037
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DOI: https://doi.org/10.1134/S0361768822020037