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 & Continuous Dynamical Systems, 2015, 35 (5) : 1969-2009. doi: 10.3934/dcds.2015.35.1969
References:
[1]

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.  Google Scholar

[2]

Celestial Mechanics and Dynamical Astronomy, 114 (2012), 319-340. doi: 10.1007/s10569-012-9417-z.  Google Scholar

[3]

Annales de l'Institut Fourier, 59 (2009), 2839-2890. doi: 10.5802/aif.2510.  Google Scholar

[4]

submited, 2012. Google Scholar

[5]

Methods Appl. Anal., 8 (2001), 113-120.  Google Scholar

[6]

Dover Publications Inc., New York, 1960.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

Celestial Mechanics and Dynamical Astronomy, 114 (2012), 319-340. doi: 10.1007/s10569-012-9417-z.  Google Scholar

[3]

Annales de l'Institut Fourier, 59 (2009), 2839-2890. doi: 10.5802/aif.2510.  Google Scholar

[4]

submited, 2012. Google Scholar

[5]

Methods Appl. Anal., 8 (2001), 113-120.  Google Scholar

[6]

Dover Publications Inc., New York, 1960.  Google Scholar

[1]

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

[2]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214

[3]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[4]

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

[5]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028

[6]

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

[7]

Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021019

[8]

Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks & Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007

[9]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2021, 13 (1) : 25-53. doi: 10.3934/jgm.2021001

[10]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3295-3317. doi: 10.3934/dcds.2020406

[11]

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

[12]

Nouressadat Touafek, Durhasan Turgut Tollu, Youssouf Akrour. On a general homogeneous three-dimensional system of difference equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021017

[13]

Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029

[14]

Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225

[15]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3319-3341. doi: 10.3934/dcds.2020407

[16]

Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021054

[17]

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

[18]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392

[19]

Monica Conti, Lorenzo Liverani, Vittorino Pata. A note on the energy transfer in coupled differential systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021042

[20]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]