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Regular and Chaotic Dynamics, 2014, Volume 19, Issue 2, Pages 145–161
DOI: https://doi.org/10.1134/S1560354714020014
(Mi rcd106)
 

This article is cited in 7 scientific papers (total in 7 papers)

Remarks on Integrable Systems

Valery V. Kozlov

Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia
Citations (7)
References:
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.
Received: 02.09.2013
Accepted: 23.09.2013
Bibliographic databases:
Document Type: Article
MSC: 34C14
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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\vol 19
\issue 2
\pages 145--161
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  • This publication is cited in the following 7 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Abstract page:347
    References:74
     
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