# American Institute of Mathematical Sciences

November  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).   Google Scholar [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.   Google Scholar [3] G. Casale, Morales-Ramis theorems via Malgrange pseudogroup,, Annales de l'Institut Fourier, 59 (2009), 2593.  doi: 10.5802/aif.2501.  Google Scholar [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.  Google Scholar [5] N. V. Grigorenko, Abelian extensions in Picard-Vessiot theory,, Mat. Zametki, 17 (1975), 113.   Google Scholar [6] J. E. Humphreys, Linear Algebraic Groups,, Graduate Texts in Mathematics, (1975).   Google Scholar [7] E. R. Kolchin, Algebraic groups and algebraic dependence,, Amer. J. Math., 90 (1968), 1151.  doi: 10.2307/2373294.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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).   Google Scholar

show all references

##### References:
 [1] M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité,, Cours Spécialisés 8, (2001).   Google Scholar [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.   Google Scholar [3] G. Casale, Morales-Ramis theorems via Malgrange pseudogroup,, Annales de l'Institut Fourier, 59 (2009), 2593.  doi: 10.5802/aif.2501.  Google Scholar [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.  Google Scholar [5] N. V. Grigorenko, Abelian extensions in Picard-Vessiot theory,, Mat. Zametki, 17 (1975), 113.   Google Scholar [6] J. E. Humphreys, Linear Algebraic Groups,, Graduate Texts in Mathematics, (1975).   Google Scholar [7] E. R. Kolchin, Algebraic groups and algebraic dependence,, Amer. J. Math., 90 (1968), 1151.  doi: 10.2307/2373294.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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).   Google Scholar
 [1] Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827 [2] Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 [3] Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 [4] Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298 [5] Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029 [6] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [7] Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166 [8] Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 [9] Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 [10] Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 [11] Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 [12] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [13] Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637 [14] Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 [15] Zhimin Chen, Kaihui Liu, Xiuxiang Liu. Evaluating vaccination effectiveness of group-specific fractional-dose strategies. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021062 [16] Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 [17] Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181 [18] Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028 [19] Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136 [20] Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185

2019 Impact Factor: 1.338

## Metrics

• PDF downloads (21)
• HTML views (0)
• Cited by (5)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]