Skip to main content
Log in

Normalization of a Periodic Hamiltonian System

  • Published:
Programming and Computer Software Aims and scope Submit manuscript

Abstract

First, the normal form in the vicinity of the stationary solution of the autonomous Hamiltonian system is recalled. Next, linear periodic Hamiltonian systems are considered. For them, normal forms of the Hamiltonians in the complex and real cases are found. A feature of the real case in the situation of parametric resonance is discovered. Next, normal forms of Hamiltonians of nonlinear periodic systems are found. Using an additional canonical transformation, such a normal form can be always reduced to an autonomous Hamiltonian system that preserves all small parameters and symmetries of the original system. The local families of fixed points of this autonomous system are associated with families of periodic solutions to the original system. A similar theory is constructed in the neighborhood of the periodic solution to the autonomous system. All transformations are algorithmic and can be implemented in a computer algebra system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Bruno, A.D., Analytical form of differential equations, Transactions Moscow Math. Soc., 1972, vol. 26, pp. 199–226.

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

  3. Zhuravlev, V.F., Petrov, A.G., and Shunderyuk, M.M., Selected Problems of Hamiltonian Mechanics, Moscow: LENAND, 2015 [in Russian].

    Google Scholar 

  4. Williamson, J., The exponential representation of canonical matrices, Am. Math. J., 1939, vol. 61, no. 4, pp. 897–911.

    Article  MathSciNet  Google Scholar 

  5. Bruno, A.D., Normal form of a periodic Hamiltonian system with \(n\) degrees of freedom, Preprint of the Keldysh Inst. of Applied Mathematics, Russ. Acad. Sci., Moscow, 2018, no. 223,https://doi.org/10.20948/prepr-2018-223. http://library.keldysh.ru/preprint.asp?id 18-223 [in Russian].

  6. Bruno, A.D., Power Geometry in Algebraic and Differential Equations, Moscow: Nauka, 1998; Amsterdam: Elsevier, 2000.

  7. Bruno, A.D., Normal form of a Hamiltonian system with periodic perturbation, Computational Mathematics and Mathematical Physics, 2020, Vol. 60, No. 1, pp. 37–53.

  8. Baider, A. and Sanders, J.A., Unique normal forms: The nilpotent Hamiltonian case, J. Differ. Equations, 1991, vol. 92, pp. 282–304.

    Article  MathSciNet  Google Scholar 

  9. Wolfram, S., The Mathematica Book, Wolfram Media, Inc., 2003.

    MATH  Google Scholar 

  10. Thompson, I., Understanding Maple, Cambridge Univ. Press, 2016.

    MATH  Google Scholar 

  11. Meurer, A., et al., SymPy: Symbolic computing in Python, Peer J. Comput. Sci., 2017, vol. 3, p. e103. https://doi.org/10.7717/peerj-cs.103

    Article  Google Scholar 

  12. Malashonok, G.I., MathPartner Computer Algebra, Program. Comput. Software, 2017, vol. 43, no. 2, pp. 112–118.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. D. Bruno.

Additional information

Translated by A. Klimontovich

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bruno, A.D. Normalization of a Periodic Hamiltonian System. Program Comput Soft 46, 76–83 (2020). https://doi.org/10.1134/S0361768820020048

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768820020048

Navigation