# 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 - A, 2015, 35 (5) : 1969-2009. doi: 10.3934/dcds.2015.35.1969
##### References:
 [1] A. Aparicio Monforte and J.-A. Weil, A reduction method for higher order variational equations of Hamiltonian systems,, In Symmetries and related topics in differential and difference equations, 549 (2011), 1. doi: 10.1090/conm/549/10850. Google Scholar [2] T. Combot, Non-integrability of the equal mass; n-body problem with non-zero angular momentum,, Celestial Mechanics and Dynamical Astronomy, 114 (2012), 319. doi: 10.1007/s10569-012-9417-z. Google Scholar [3] 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. Google Scholar [4] G. Duval and A. J. Maciejewski, Integrability of Homogeneous potential of degree $k = \pm 2$. An application of higher variational equations,, submited, (2012). Google Scholar [5] J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential,, Methods Appl. Anal., 8 (2001), 113. Google Scholar [6] E. G. C. Poole, Introduction to the Theory of Linear Differential Equations,, Dover Publications Inc., (1960). Google Scholar

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##### References:
 [1] A. Aparicio Monforte and J.-A. Weil, A reduction method for higher order variational equations of Hamiltonian systems,, In Symmetries and related topics in differential and difference equations, 549 (2011), 1. doi: 10.1090/conm/549/10850. Google Scholar [2] T. Combot, Non-integrability of the equal mass; n-body problem with non-zero angular momentum,, Celestial Mechanics and Dynamical Astronomy, 114 (2012), 319. doi: 10.1007/s10569-012-9417-z. Google Scholar [3] 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. Google Scholar [4] G. Duval and A. J. Maciejewski, Integrability of Homogeneous potential of degree $k = \pm 2$. An application of higher variational equations,, submited, (2012). Google Scholar [5] J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential,, Methods Appl. Anal., 8 (2001), 113. Google Scholar [6] E. G. C. Poole, Introduction to the Theory of Linear Differential Equations,, Dover Publications Inc., (1960). Google Scholar
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