\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations

Abstract Related Papers Cited by
  • The present work is the first one of two papers, in which we analyse systems of higher order variational equations associated to natural Hamiltonian systems with homogeneous potential of degree $k\in\mathbb{Z}\setminus \{-1,0,1\}$. Our attempt is to give necessary conditions for complete integrability which can be deduced in a framework of differential Galois theory. We show that the higher variational equations $\mathrm{VE}_p$ of order $p\geq 2$, although complicated, have a very particular algebraic structure. More precisely, we show that if $\mathrm{VE}_1$ has virtually Abelian differential Galois group (DGG), then $\mathrm{VE}_{p}$ are solvable for an arbitrary $p>1$. We proved this inductively using what we call the second level integrals. Then we formulate the necessary and sufficient conditions in terms of these second level integrals for $\mathrm{VE}_{p}$ to be virtually Abelian. We apply the above conditions to potentials of degree $k=\pm 2$ considering their $\mathrm{VE}_p$ with $p>1$ along Darboux points. For $k= 2$, $\mathrm{VE}_1$ does not give any obstruction to the integrability. We show that under certain non-resonance condition, the only degree two integrable potential is the multidimensional harmonic oscillator. In contrast, for degree $k=-2$ potentials, all the $\mathrm{VE}_{p}$ along Darboux points are virtually Abelian.
    Mathematics Subject Classification: Primary: 37J30, 70H07, 37J35, 34M35.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité, Cours Spécialisés 8, Collection SMF, SMF et EDP Sciences, Paris, 2001.

    [2]

    A. Baider, R. C. Churchill, D. L. Rod and M. F. Singer, On the infinitesimal geometry of integrable systems, in Mechanics Day (Waterloo, ON, 1992), vol. 7 of Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 1996, 5-56.

    [3]

    G. Casale, Morales-Ramis theorems via Malgrange pseudogroup, Annales de l'Institut Fourier, 59 (2009), 2593-2610.doi: 10.5802/aif.2501.

    [4]

    G. Duval and A. J. Maciejewski, Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials, Annales de l'Institut Fourier, 59 (2009), 2839-2890.doi: 10.5802/aif.2510.

    [5]

    N. V. Grigorenko, Abelian extensions in Picard-Vessiot theory, Mat. Zametki, 17 (1975), 113-117.

    [6]

    J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1975.

    [7]

    E. R. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math., 90 (1968), 1151-1164.doi: 10.2307/2373294.

    [8]

    A. J. Maciejewski and M. Przybylska, Differential Galois theory and integrability, Internat. J. Geom. Methods in Modern Phys., 6 (2009), 1357-1390.doi: 10.1142/S0219887809004272.

    [9]

    J. J. Morales-Ruiz and J.-P. Ramis, Integrability of dynamical systems through differential Galois theory: A practical guide, in Differential algebra, complex analysis and orthogonal polynomials, Contemp. Math., Amer. Math. Soc., Providence, RI, 509 (2010), 143-220.doi: 10.1090/conm/509/09980.

    [10]

    J. J. Morales-Ruiz, J. P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. Éc. Norm. Supér, 40 (2007), 845-884.doi: 10.1016/j.ansens.2007.09.002.

    [11]

    M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, vol. 328 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2003.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(58) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return