# American Institute of Mathematical Sciences

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

 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.
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 - A, 2014, 34 (11) : 4589-4615. doi: 10.3934/dcds.2014.34.4589
##### References:
 [1] M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité,, Cours Spécialisés 8, (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, (1992), 5. [3] G. Casale, Morales-Ramis theorems via Malgrange pseudogroup,, Annales de l'Institut Fourier, 59 (2009), 2593. 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. doi: 10.5802/aif.2510. [5] N. V. Grigorenko, Abelian extensions in Picard-Vessiot theory,, Mat. Zametki, 17 (1975), 113. [6] J. E. Humphreys, Linear Algebraic Groups,, Graduate Texts in Mathematics, (1975). [7] E. R. Kolchin, Algebraic groups and algebraic dependence,, Amer. J. Math., 90 (1968), 1151. 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. 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, 509 (2010), 143. 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. 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, (2003).

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##### References:
 [1] M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité,, Cours Spécialisés 8, (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, (1992), 5. [3] G. Casale, Morales-Ramis theorems via Malgrange pseudogroup,, Annales de l'Institut Fourier, 59 (2009), 2593. 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. doi: 10.5802/aif.2510. [5] N. V. Grigorenko, Abelian extensions in Picard-Vessiot theory,, Mat. Zametki, 17 (1975), 113. [6] J. E. Humphreys, Linear Algebraic Groups,, Graduate Texts in Mathematics, (1975). [7] E. R. Kolchin, Algebraic groups and algebraic dependence,, Amer. J. Math., 90 (1968), 1151. 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. 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, 509 (2010), 143. 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. 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, (2003).
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