Tensor invariants and integration of differential equations
V. V. Kozlov
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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.
tensors, invariant forms and fields, flags, quasi-homogeneous systems, Kovalevskaya exponents, Goryachev–Chaplygin case.
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Russian Mathematical Surveys, 2019, 74:1, 111–140
MSC: Primary 58J70; Secondary 34A34, 70H05
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
\paper Tensor invariants and integration of differential equations
\jour Uspekhi Mat. Nauk
\jour Russian Math. Surveys
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