DCDS
Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations
Guillaume Duval Andrzej J. Maciejewski
Discrete & Continuous Dynamical Systems - A 2014, 34(11): 4589-4615 doi: 10.3934/dcds.2014.34.4589
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
DCDS
Integrability of potentials of degree $k \neq \pm 2$. Second order variational equations between Kolchin solvability and Abelianity
Guillaume Duval Andrzej J. Maciejewski
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 1969-2009 doi: 10.3934/dcds.2015.35.1969
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 $\mathrm{VE}_2$.
keywords: differential Galois group Hamiltonian systems higher order variational equations. Integrability homogeneous potentials

Year of publication

Related Authors

Related Keywords

[Back to Top]