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

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

Document Type: Article
UDC: 517.9+531.01
MSC: Primary 58J70; Secondary 34A34, 70H05

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
\Bibitem{Koz19} \by V.~V.~Kozlov \paper Tensor invariants and integration of differential equations \jour Uspekhi Mat. Nauk \yr 2019 \vol 74 \issue 1(445) \pages 117--148 \mathnet{http://mi.mathnet.ru/umn9866} \crossref{https://doi.org/10.4213/rm9866} \elib{http://elibrary.ru/item.asp?id=37089794} \transl \jour Russian Math. Surveys \yr 2019 \vol 74 \issue 1 \pages 111--140 \crossref{https://doi.org/10.1070/RM9866}