# American Institute of Mathematical Sciences

May  2015, 35(5): 1969-2009. doi: 10.3934/dcds.2015.35.1969

## Integrability of potentials of degree $k \neq \pm 2$. Second order variational equations between Kolchin solvability and Abelianity

 1 Laboratoire de Mathématiques et d'Informatique (LMI), INSA de Rouen, Avenue de l'Université, 76 801 Saint Etienne du Rouvray Cedex 2 Institute of Astronomy, University of Zielona Góra, Licealna 9, PL-65-417, Zielona Góra, Poland

Received  January 2013 Revised  September 2014 Published  December 2014

In our previous paper [4], we tried to extract some particular structures of the higher variational equations (the $\mathrm{VE}_p$ for $p \geq 2$), along particular solutions of natural Hamiltonian systems with homogeneous potential of degree $k=\pm 2$. We investigate these variational equations in a framework of differential Galois theory. Our aim was to obtain new obstructions for complete integrability. In this paper we extend results of [4] to the complementary cases, when the homogeneous potential has integer degree of homogeneity $k\in\mathbb{Z}$, and $|k| \geq 3$. Since these cases are much more general and complicated, we restrict our study only to the second order variational equation $\mathrm{VE}_2$.
Citation: Guillaume Duval, Andrzej J. Maciejewski. Integrability of potentials of degree $k \neq \pm 2$. Second order variational equations between Kolchin solvability and Abelianity. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1969-2009. doi: 10.3934/dcds.2015.35.1969
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##### References:
 [1] In Symmetries and related topics in differential and difference equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 549 (2011), 1-15. doi: 10.1090/conm/549/10850.  Google Scholar [2] Celestial Mechanics and Dynamical Astronomy, 114 (2012), 319-340. doi: 10.1007/s10569-012-9417-z.  Google Scholar [3] Annales de l'Institut Fourier, 59 (2009), 2839-2890. doi: 10.5802/aif.2510.  Google Scholar [4] submited, 2012. Google Scholar [5] Methods Appl. Anal., 8 (2001), 113-120.  Google Scholar [6] Dover Publications Inc., New York, 1960.  Google Scholar
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