Invariant manifolds and periodic solutions of three degrees of freedom Hamiltonian systems

  • F. Verhulst
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 195)


Hamiltonian systems considered near a stable equilibrium point can be analyzed using normalization techniques à la Birkhoff or, equivalently, averaging in one of its canonical forms. It is well known that two degrees of freedom systems become integrable upon normalization, the integrals being asymptotic integrals (valid for all time) for the original system. In the case of three degrees of freedom the situation is more complex: there are a number of results concerning integrability and there are many open problems. Both in two and three degrees of freedom, the periodic solutions admit systematic analysis although the complexity increases enormously with the dimension.

Most results are concerned with the generic cases but, keeping an eye on applications, we also have to allow for degeneracies and bifurcations arising from certain symmetry properties. As an illustration of some of the mathematical theory we shall consider applications in the theory of vibrations and in astrophysics.


Periodic Solution Periodic Orbit Hamiltonian System Invariant Manifold Discrete Symmetry 
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  1. [1]
    F. Verhulst, Asymptotic analysis of Hamiltonian systems, Lecture Notes Mathematics 985 (F. Verhulst, ed.) Springer-Verlag (1983).Google Scholar
  2. [2]
    J.A. Sanders and F. Verhulst, Averaging methods in nonlinear dynamical systems, Appl. Math. Sciences, Springer-Verlag (1984).Google Scholar
  3. [3]
    M.A. Lieberman, Arnold diffusion in Hamiltonian systems with three degrees of freedom; in Nonlinear Dynamics (R. Helleman ed.) p. 119–142 (1982), New York Academy of Sciences, New York.Google Scholar
  4. [4]
    E. van der Aa and J.A. Sanders, The 1: 2: 1-resonance, its periodic orbits and integrals, Lecture Notes Math. 711 (F. Verhulst, ed.), Springer-Verlag (1979).Google Scholar
  5. [5]
    E. van der Aa, First-order resonances in three-degrees-of-freedom systems, prepr. 197, Math. Inst. Rijksuniversiteit Utrecht (1981), to be publ. in Celestial Mechanics (1983).Google Scholar
  6. [6]
    J.J. Duistermaat, Non-integrability of the 1:1:2-resonance, prepr. 281, Math. Inst., Rijksuniversiteit Utrecht (1983).Google Scholar
  7. [7]
    F. Verhulst, Discrete-symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies, Phil. Trans. roy. Soc. London A, 290, 435–465 (1979).ADSCrossRefMATHGoogle Scholar
  8. [8]
    T. de Zeeuw, Periodic orbits in triaxial galaxies, Proc. CECAM Workshop on Structure, Formation and Evolution of Galaxies (J. Audouze and C. Norman, eds.) page 11, Paris (1982).Google Scholar
  9. [9]
    E.A. Jackson, Nonlinearity and irreversability in lattice dynamics, Rocky Mount. J. Math. 8, 127–196 (1978).CrossRefGoogle Scholar
  10. [10]
    D. Merritt and T. de Zeeuw, Orbital configurations for gas in elliptical galaxies, Ap. J. Letters (1983).Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • F. Verhulst
    • 1
  1. 1.Mathematisch InstituutRijksuniversiteit UtrechtTA UtrechtThe Netherlands

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