American Institute of Mathematical Sciences

Advanced
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

Guillaume Duval 1, and  Andrzej J. Maciejewski 2,

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.
Keywords: Integrability, homogeneous potentials, differential Galois group, higher order variational equations., Hamiltonian systems.
Mathematics Subject Classification: Primary: 37J30, 70H07, 37J35, 34M3.
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, 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, Collection SMF, SMF et EDP Sciences, Paris, 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, ON, 1992), vol. 7 of Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 1996, 5-56.  Google Scholar

[3]

G. Casale, Morales-Ramis theorems via Malgrange pseudogroup, Annales de l'Institut Fourier, 59 (2009), 2593-2610. 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), 2839-2890. doi: 10.5802/aif.2510.  Google Scholar

[5]

N. V. Grigorenko, Abelian extensions in Picard-Vessiot theory, Mat. Zametki, 17 (1975), 113-117.  Google Scholar

[6]

J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1975.  Google Scholar

[7]

E. R. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math., 90 (1968), 1151-1164. 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), 1357-1390. 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, complex analysis and orthogonal polynomials, Contemp. Math., Amer. Math. Soc., Providence, RI, 509 (2010), 143-220. 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), 845-884. 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, Berlin, 2003.  Google Scholar

show all references

References:
[1]

M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité, Cours Spécialisés 8, Collection SMF, SMF et EDP Sciences, Paris, 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, ON, 1992), vol. 7 of Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 1996, 5-56.  Google Scholar

[3]

G. Casale, Morales-Ramis theorems via Malgrange pseudogroup, Annales de l'Institut Fourier, 59 (2009), 2593-2610. 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), 2839-2890. doi: 10.5802/aif.2510.  Google Scholar

[5]

N. V. Grigorenko, Abelian extensions in Picard-Vessiot theory, Mat. Zametki, 17 (1975), 113-117.  Google Scholar

[6]

J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1975.  Google Scholar

[7]

E. R. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math., 90 (1968), 1151-1164. 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), 1357-1390. 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, complex analysis and orthogonal polynomials, Contemp. Math., Amer. Math. Soc., Providence, RI, 509 (2010), 143-220. 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), 845-884. 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, Berlin, 2003.  Google Scholar

[1]

Jaume Llibre, Yuzhou Tian. Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021228
[2]

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
[3]

Regina Martínez, Carles Simó. Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 1-24. doi: 10.3934/dcds.2011.29.1
[4]

Mitsuru Shibayama. Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3707-3719. doi: 10.3934/dcds.2015.35.3707
[5]

Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227
[6]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
[7]

Dung Le. Higher integrability for gradients of solutions to degenerate parabolic systems. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 597-608. doi: 10.3934/dcds.2010.26.597
[8]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827
[9]

Baruch Cahlon. Sufficient conditions for oscillations of higher order neutral delay differential equations. Conference Publications, 1998, 1998 (Special) : 124-137. doi: 10.3934/proc.1998.1998.124
[10]

R.S. Dahiya, A. Zafer. Oscillation theorems of higher order neutral type differential equations. Conference Publications, 1998, 1998 (Special) : 203-219. doi: 10.3934/proc.1998.1998.203
[11]

Peiguang Wang, Xiran Wu, Huina Liu. Higher order convergence for a class of set differential equations with initial conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3233-3248. doi: 10.3934/dcdss.2020342
[12]

Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
[13]

Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557
[14]

Jaume Llibre, Claudia Valls. Analytic integrability of a class of planar polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2657-2661. doi: 10.3934/dcdsb.2015.20.2657
[15]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81
[16]

Chiara Leone, Anna Verde, Giovanni Pisante. Higher integrability results for non smooth parabolic systems: The subquadratic case. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 177-190. doi: 10.3934/dcdsb.2009.11.177
[17]

Kristian Moring, Christoph Scheven, Sebastian Schwarzacher, Thomas Singer. Global higher integrability of weak solutions of porous medium systems. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1697-1745. doi: 10.3934/cpaa.2020069
[18]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[19]

Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933
[20]

Yu Guo, Xiao-Bao Shu, Qianbao Yin. Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021236

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy