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Russian Mathematical Surveys, 2019, Volume 74, Issue 1, Pages 111–140
DOI: https://doi.org/10.1070/RM9866
(Mi rm9866)
 

This article is cited in 45 scientific papers (total in 45 papers)

Tensor invariants and integration of differential equations

V. V. Kozlov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
References:
Abstract: The connection between tensor invariants of systems of differential equations and explicit integration of them is discussed. A general result on the integrability of dynamical systems admitting a complete set of integral invariants in the sense of Cartan is proved. The existence of an invariant 1-form is related to the representability of the dynamical system in Hamiltonian form (with a symplectic structure which may be degenerate). This general idea is illustrated using an example of linear systems of differential equations. A general concept of flags of tensor invariants is introduced. General relations between the Kovalevskaya exponents of quasi-homogeneous systems of differential equations and flags of quasi-homogeneous tensor invariants having a certain structure are established. Results of a general nature are applied, in particular, to show that the general solution of the equations of rotation for a rigid body is branching in the Goryachev–Chaplygin case.
Bibliography: 50 titles.
Keywords: tensors, invariant forms and fields, flags, quasi-homogeneous systems, Kovalevskaya exponents, Goryachev–Chaplygin case.
Received: 30.11.2018
Russian version:
Uspekhi Matematicheskikh Nauk, 2019, Volume 74, Issue 1(445), Pages 117–148
DOI: https://doi.org/10.4213/rm9866
Bibliographic databases:
Document Type: Article
UDC: 517.9+531.01
MSC: Primary 58J70; Secondary 34A34, 70H05
Language: English
Original paper language: Russian
Citation: V. V. Kozlov, “Tensor invariants and integration of differential equations”, Uspekhi Mat. Nauk, 74:1(445) (2019), 117–148; Russian Math. Surveys, 74:1 (2019), 111–140
Citation in format AMSBIB
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  • https://doi.org/10.1070/RM9866
  • https://www.mathnet.ru/eng/rm/v74/i1/p117
  • This publication is cited in the following 45 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Успехи математических наук Russian Mathematical Surveys
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    Abstract page:1053
    Russian version PDF:524
    English version PDF:66
    References:111
    First page:99
     
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