-
Previous Article
Blow-up set for a superlinear heat equation and pointedness of the initial data
- DCDS Home
- This Issue
-
Next Article
Delay-dependent stability criteria for neutral delay differential and difference equations
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 |
References:
[1] |
M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité, Cours Spécialisés 8, Collection SMF, SMF et EDP Sciences, Paris, 2001. |
[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. |
[3] |
G. Casale, Morales-Ramis theorems via Malgrange pseudogroup, Annales de l'Institut Fourier, 59 (2009), 2593-2610.
doi: 10.5802/aif.2501. |
[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. |
[5] |
N. V. Grigorenko, Abelian extensions in Picard-Vessiot theory, Mat. Zametki, 17 (1975), 113-117. |
[6] |
J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1975. |
[7] |
E. R. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math., 90 (1968), 1151-1164.
doi: 10.2307/2373294. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité, Cours Spécialisés 8, Collection SMF, SMF et EDP Sciences, Paris, 2001. |
[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. |
[3] |
G. Casale, Morales-Ramis theorems via Malgrange pseudogroup, Annales de l'Institut Fourier, 59 (2009), 2593-2610.
doi: 10.5802/aif.2501. |
[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. |
[5] |
N. V. Grigorenko, Abelian extensions in Picard-Vessiot theory, Mat. Zametki, 17 (1975), 113-117. |
[6] |
J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1975. |
[7] |
E. R. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math., 90 (1968), 1151-1164.
doi: 10.2307/2373294. |
[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. |
[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. |
[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. |
[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. |
[1] |
Jaume Llibre, Yuzhou Tian. Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4305-4316. 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2010, 26 (2) : 597-608. doi: 10.3934/dcds.2010.26.597 |
[8] |
Sergi Simon. Linearised higher variational equations. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2022, 27 (8) : 4455-4471. doi: 10.3934/dcdsb.2021236 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]