Matrix-second order differential equations and chaotic Hamiltonian systems

  • A. Celletti
  • J. P. Francoise
Brief Reports


We consider matrix-second order differential equations which are perturbations of the harmonic flow on the space of matrices. Experimental evidence of the non integrability of the two degrees of freedom Hamiltonian system provides an indication of the non existence of a Lax pair with commuting eigenvalues for perturbations of order six. This shows the specificity of quartic perturbations for which such a Lax pair was precedently obtained.


Differential Equation Experimental Evidence Mathematical Method Hamiltonian System Order Differential Equation 
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  1. [C]
    F. Calogero,Solution of the one-dimensional n-body problems with quadratic and/or inversely quadratic pair potentials. J. of Math. Phys.12, 419 (1973).Google Scholar
  2. [F-W-M]
    A. P. Fordy, S. Wojciechowski and I. Marshall,A family of integrable quartic potentials related to symmetric spaces. Phys. Lett.1134, 395 (1986).Google Scholar
  3. [F-R]
    J.-P. Francoise and O. Ragnisco,Matrix second-order differential equations and Hamiltonian systems of quartic type. To appear inAnnales de l'Institut H. Poincaré.Google Scholar
  4. [H-H]
    M. Hénon and C. Heiles,The applicability of the third integral of motion: some numerical experiments. Astron. J.69, 73 (1964).Google Scholar
  5. [K-K-S]
    J. Kazhdan, B. Kostant and S. Sternberg,Hamiltonian group actions and dynamical systems of Calogero type. Comm. Pure Appl. Math.31, 481 (1978).Google Scholar
  6. [M]
    J. Moser,Various aspects of integrable Hamiltonian systems. Proc. CIME conf. held in Bressanone, 1978.Google Scholar
  7. [S]
    C. Scovel,Private communication.Google Scholar
  8. [Y]
    H. Yoshida,Non-integrability of the truncated Toda lattice Hamiltonian at any order. Comm. Math. Phys.116, 529 (1988).Google Scholar
  9. [Z]
    S. L. Ziglin,Branching of solutions and the nonexistence of first integrals in Hamiltonian mechanics. Funct. Anal. Appl.16, 181 (1983).Google Scholar

Copyright information

© Birkhäuser Verlag 1989

Authors and Affiliations

  • A. Celletti
    • 1
  • J. P. Francoise
    • 2
  1. 1.Forschungsinstitut für MathematikETH-ZentrumZürich
  2. 2.Bat. n.425, MathématiquesUniversité de Paris XIOrsay, CedexFrance

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