Tensor invariants and integration of differential equations

V. V. Kozlov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 1, pp. 111-140

Résumé

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.

Détail

Keywords: tensors, invariant forms and fields, flags, quasi-homogeneous systems, Kovalevskaya exponents, Goryachev–Chaplygin case.

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