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Uspekhi Mat. Nauk, 2019, Volume 74, Issue 1(445), Pages 117–148 (Mi umn9866)  

This article is cited in 4 scientific papers (total in 4 papers)

Tensor invariants and integration of differential equations

V. V. Kozlov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

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.

DOI: https://doi.org/10.4213/rm9866

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English version:
Russian Mathematical Surveys, 2019, 74:1, 111–140

Bibliographic databases:

UDC: 517.9+531.01
MSC: Primary 58J70; Secondary 34A34, 70H05
Received: 30.11.2018

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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    This publication is cited in the following articles:
    1. V. I. Bogachev, “Non-uniform Kozlov–Treschev averagings in the ergodic theorem”, Russian Math. Surveys, 75:3 (2020), 393–425  mathnet  crossref  crossref  mathscinet  isi  elib
    2. A. T. Il'ichev, “Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes”, Russian Math. Surveys, 75:5 (2020), 843–882  mathnet  crossref  crossref  mathscinet  isi  elib
    3. A. V. Borisov, A. V. Tsiganov, “Chaplygin ball in a solenoidal field”, Russian Math. Surveys, 76:3 (2021), 546–548  mathnet  crossref  crossref  isi  elib
    4. V. M. Buchstaber, A. V. Mikhailov, “Integrable polynomial Hamiltonian systems and symmetric powers of plane algebraic curves”, Russian Math. Surveys, 76:4 (2021), 587–652  mathnet  crossref  crossref  isi
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