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 Regul. Chaotic Dyn., 2014, Volume 19, Issue 2, Pages 145–161 (Mi rcd106)

Remarks on Integrable Systems

Valery V. Kozlov

Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

Abstract: The problem of integrability conditions for systems of differential equations is discussed. Darboux's classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an $n$-dimensional space, which admit the algebra of symmetry fields of dimension $\geqslant n$. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.

Keywords: integrability by quadratures, adjoint system, Hamiltonian equations, Euler–Jacobi theorem, Lie theorem, symmetries

DOI: https://doi.org/10.1134/S1560354714020014

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Document Type: Article
MSC: 34C14
Accepted:23.09.2013
Language: English

Citation: Valery V. Kozlov, “Remarks on Integrable Systems”, Regul. Chaotic Dyn., 19:2 (2014), 145–161

Citation in format AMSBIB
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\paper Remarks on Integrable Systems
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\pages 145--161
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Valery V. Kozlov, “The Dynamics of Systems with Servoconstraints. II”, Regul. Chaotic Dyn., 20:4 (2015), 401–427
2. Vladimir Dragović, Borislav Gajić, Božidar Jovanović, “Note on Free Symmetric Rigid Body Motion”, Regul. Chaotic Dyn., 20:3 (2015), 293–308
3. V. Kozlov, “The phenomenon of reversal in the Euler–Poincaré–Suslov nonholonomic systems”, J. Dyn. Control Syst., 22:4 (2016), 713–724
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