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Normal Form of a Hamiltonian System with a Periodic Perturbation

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Abstract

A perturbed Hamiltonian system with a time-independent unperturbed part and a time-periodic perturbation is considered near a stationary solution. First, the normal form of an autonomous Hamiltonian function is recalled. Then the normal form of a periodic perturbation is described. This form can always be reduced to an autonomous Hamiltonian function, which makes it possible to compute local families of periodic solutions of the original system. First approximations of some of these families are found by computing the Newton polyhedron of the reduced normal form of the Hamiltonian function. Computer algebra problems arising in these computations are briefly discussed.

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REFERENCES

  1. A. D. Bruno, “Normal form of the periodic Hamiltonian system with n degrees of freedom,” Preprint No. 223, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2018). https://doi.org/10.20948/prepr-2018-22

  2. A. D. Bruno, The Restricted 3-Body Problem: Plane Periodic Orbits (Nauka, Moscow, 1990; Walter de Gruyter, Berlin, 1994).

  3. A. D. Bruno, Power Geometry in Algebraic and Differential Equations (Fizmatlit, Moscow, 1998; Elsevier Science, Amsterdam, 2000).

  4. V. F. Zhuravlev, A. G. Petrov, and M. M. Shunderyuk, Selected Problems in Hamiltonian Mechanics (LENAND, Moscow, 2015) [in Russian].

    Google Scholar 

  5. K. Fukuda, “Exact algorithms and software in optimization and polyhedral computation,” Proceedings of ISSAC’08 21st International Symposium on Symbolic and Algebraic Computations (ACM, New York, 2008), pp. 333–334.

  6. C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Software 22 (4), 469–483 (1996). http://www.qhull.org.

    Article  MathSciNet  Google Scholar 

  7. A. P. Markeev, “The rotational motion of dynamically symmetric satellite in an elliptical orbit,” Cosmic Res. 5 (4), 457–465 (1967).

    Google Scholar 

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Authors and Affiliations

  1. Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

    A. D. Bruno

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  1. A. D. Bruno
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Correspondence to A. D. Bruno.

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Translated by I. Ruzanova

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Bruno, A.D. Normal Form of a Hamiltonian System with a Periodic Perturbation. Comput. Math. and Math. Phys. 60, 36–52 (2020). https://doi.org/10.1134/S0965542520010066

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  • DOI: https://doi.org/10.1134/S0965542520010066

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