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}
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https://www.mathnet.ru/eng/rcd553
https://www.mathnet.ru/eng/rcd/v14/i3/p323
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