Analysis of a Constrained Two-Body Problem

  • Wojciech SzumińskiEmail author
  • Tomasz Stachowiak
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 182)


We consider the system of two material points that interact by elastic forces according to Hooke’s law and their motion is restricted to certain curves lying on the plane. The nonintegrability of this system and idea of the proof are communicated. Moreover, the analysis of global dynamics by means of Poincaré cross sections is given and local analysis in the neighborhood of an equilibrium is performed by applying the Birkhoff normal form. Conditions of linear stability are determined and some particular periodic solutions are identified.



The work has been supported by grants No. DEC-2013/09/B/ST1/ 04130 and DEC-2011/02/A/ST1/00208 of National Science Centre of Poland.


  1. 1.
    Birkhoff, G. D.: Dynamical Systems. American Math Society (1927)Google Scholar
  2. 2.
    Jorba, Á.: A Methodology for the Numerical Computation of Normal Forms, Centre Manifolds and First Integrals of Hamiltonian Systems. Experimental Mathematics. 8, 155–195 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Markeev, A. P.: Libration Points in Celestial Mechanics and Cosmodynamics, Nauka, Moscow (1978), In RussianGoogle Scholar
  4. 4.
    Merkin, D. R.: Introduction to the Theory of Stability, Texts in Applied Mathematics, 24, Springer-Verlag, New York (1997)Google Scholar
  5. 5.
    Morales-Ruiz, J. J.: Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Mathematics, Birkhauser Verlag, Basel (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Perez-Marco, R.: Convergence or generic divergence of the Birkhoff normal form. Ann. of Math. 157, 557–574 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Szumiński, W., Przybylska, M.: Non-integrability of constrained n-body problems with Newton and Hooke interactions. Work in progressGoogle Scholar
  8. 8.
    Van der Put, M., Singer, M. F.: Galois theory of linear differential equations, Springer-Verlag, Berlin (2003)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Physics, University of Zielona GóraZielona GóraPoland
  2. 2.Center for Theoretical Physics, Polish Academy of SciencesWarsawPoland

Personalised recommendations