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November  2014, 34(11): 4589-4615. doi: 10.3934/dcds.2014.34.4589

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

Guillaume Duval 1, and  Andrzej J. Maciejewski 2,

1. 

Laboratoire de Mathématiques et d'Informatique (LMI), INSA de Rouen, Avenue de l'Université, 76 801 Saint Etienne du Rouvray Cedex, France
2. 

Kepler Institute of Astronomy, University of Zielona Góra, Licealna 9, PL-65-417, Zielona Góra, Poland

Received  January 2013 Revised  March 2014 Published  May 2014

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.
Keywords: Integrability, homogeneous potentials, differential Galois group, higher order variational equations., Hamiltonian systems.
Mathematics Subject Classification: Primary: 37J30, 70H07, 37J35, 34M3.
Citation: Guillaume Duval, Andrzej J. Maciejewski. Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4589-4615. doi: 10.3934/dcds.2014.34.4589
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show all references

References:
[1]

Cours Spécialisés 8, Collection SMF, SMF et EDP Sciences, Paris, 2001.  Google Scholar

[2]

in Mechanics Day (Waterloo, ON, 1992), vol. 7 of Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 1996, 5-56.  Google Scholar

[3]

Annales de l'Institut Fourier, 59 (2009), 2593-2610. doi: 10.5802/aif.2501.  Google Scholar

[4]

Annales de l'Institut Fourier, 59 (2009), 2839-2890. doi: 10.5802/aif.2510.  Google Scholar

[5]

Mat. Zametki, 17 (1975), 113-117.  Google Scholar

[6]

Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1975.  Google Scholar

[7]

Amer. J. Math., 90 (1968), 1151-1164. doi: 10.2307/2373294.  Google Scholar

[8]

Internat. J. Geom. Methods in Modern Phys., 6 (2009), 1357-1390. doi: 10.1142/S0219887809004272.  Google Scholar

[9]

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.  Google Scholar

[10]

Ann. Sci. Éc. Norm. Supér, 40 (2007), 845-884. doi: 10.1016/j.ansens.2007.09.002.  Google Scholar

[11]

Springer-Verlag, Berlin, 2003.  Google Scholar

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