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 and Continuous Dynamical Systems, 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, Contemp. Math., Amer. Math. Soc., Providence, RI, 549 (2011), 1-15. doi: 10.1090/conm/549/10850.

[2]

T. Combot, Non-integrability of the equal mass; n-body problem with non-zero angular momentum, Celestial Mechanics and Dynamical Astronomy, 114 (2012), 319-340. doi: 10.1007/s10569-012-9417-z.

[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), 2839-2890. doi: 10.5802/aif.2510.

[4]

G. Duval and A. J. Maciejewski, Integrability of Homogeneous potential of degree $k = \pm 2$. An application of higher variational equations, submited, 2012.

[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), 113-120.

[6]

E. G. C. Poole, Introduction to the Theory of Linear Differential Equations, Dover Publications Inc., New York, 1960.

show all references

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, Contemp. Math., Amer. Math. Soc., Providence, RI, 549 (2011), 1-15. doi: 10.1090/conm/549/10850.

[2]

T. Combot, Non-integrability of the equal mass; n-body problem with non-zero angular momentum, Celestial Mechanics and Dynamical Astronomy, 114 (2012), 319-340. doi: 10.1007/s10569-012-9417-z.

[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), 2839-2890. doi: 10.5802/aif.2510.

[4]

G. Duval and A. J. Maciejewski, Integrability of Homogeneous potential of degree $k = \pm 2$. An application of higher variational equations, submited, 2012.

[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), 113-120.

[6]

E. G. C. Poole, Introduction to the Theory of Linear Differential Equations, Dover Publications Inc., New York, 1960.

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