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Nelin. Dinam., 2013, Volume 9, Number 3, Pages 459–478 (Mi nd400)  

This article is cited in 1 scientific paper (total in 1 paper)

Notes on integrable systems

Valery V. Kozlov

Steklov Mathematical Institute of the Russian Academy of Sciences, Gubkina st. 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 permit the algebra of symmetry fields of dimension $\ge n$. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momentums 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, Hamilton equations, Euler–Jacobi theorem, Lie theorem, symmetries.

Full text: PDF file (373 kB)
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Document Type: Article
UDC: 517.9
MSC: 34C14
Received: 02.09.2013
Revised: 23.09.2013

Citation: Valery V. Kozlov, “Notes on integrable systems”, Nelin. Dinam., 9:3 (2013), 459–478

Citation in format AMSBIB
\Bibitem{Koz13}
\by Valery~V.~Kozlov
\paper Notes on integrable systems
\jour Nelin. Dinam.
\yr 2013
\vol 9
\issue 3
\pages 459--478
\mathnet{http://mi.mathnet.ru/nd400}


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    This publication is cited in the following articles:
    1. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dinamika negolonomnykh sistem, sostoyaschikh iz sfericheskoi obolochki s podvizhnym tverdym telom vnutri”, Nelineinaya dinam., 9:3 (2013), 547–566  mathnet
  • Нелинейная динамика
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