Abstract:
This paper deals with non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to deal with these problems. We display a family of Hamiltonian systems which require the use of order k variational equations, for arbitrary values of $k$, to prove non-integrability. Moreover, using third order variational equations we prove the non-integrability of a non-linear springpendulum problem for the values of the parameter that can not be decided using first order variational equations.
Citation:
R. Martínez, C. Simó, “Non-Integrability of Hamiltonian Systems Through High Order Variational Equations: Summary of Results and Examples”, Regul. Chaotic Dyn., 14:3 (2009), 323–348
\Bibitem{MarSim09}
\by R. Mart{\'\i}nez, C. Sim\'o
\paper Non-Integrability of Hamiltonian Systems Through High Order Variational Equations: Summary of Results and Examples
\jour Regul. Chaotic Dyn.
\yr 2009
\vol 14
\issue 3
\pages 323--348
\mathnet{http://mi.mathnet.ru/rcd553}
\crossref{https://doi.org/10.1134/S1560354709030010}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2525618}
\zmath{https://zbmath.org/?q=an:1229.37058}
Linking options:
https://www.mathnet.ru/eng/rcd553
https://www.mathnet.ru/eng/rcd/v14/i3/p323
This publication is cited in the following 10 articles:
Joan Gimeno, Àngel Jorba, Marc Jorba-Cuscó, Narcís Miguel, Maorong Zou, “Numerical integration of high-order variational equations of ODEs”, Applied Mathematics and Computation, 442 (2023), 127743
Sergi Simon, “Conditions and evidence for non-integrability in the Friedmann-Robertson-Walker Hamiltonian”, JNMP, 21:1 (2021), 1
Tatyana E. Churkina, Sergey Y. Stepanov, “On the Stability of Periodic Mercury-type Rotations”, Regul. Chaotic Dyn., 22:7 (2017), 851–864
A. Aparicio-Monforte, T. Dreyfus, J.-A. Weil, “Liouville integrability: An effective Morales–Ramis–Simó theorem”, Journal of Symbolic Computation, 74 (2016), 537
Juan J. Morales-Ruiz, “Picard–Vessiot theory and integrability”, Journal of Geometry and Physics, 87 (2015), 314
Carles Simó, “Measuring the total amount of chaos in some Hamiltonian systems”, Discrete & Continuous Dynamical Systems - A, 34:12 (2014), 5135
Thierry Combot, Christoph Koutschan, “Third order integrability conditions for homogeneous potentials of degree -1”, Journal of Mathematical Physics, 53:8 (2012)
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 - A, 29:1 (2011), 1
O. Pujol, J.P. Pérez, J.P. Ramis, C. Simó, S. Simon, J.A. Weil, “Swinging Atwood Machine: Experimental and numerical results, and a theoretical study”, Physica D: Nonlinear Phenomena, 239:12 (2010), 1067
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