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.
This publication is cited in the following 51 articles:
M. V. Shamolin, “Invarianty odnorodnykh dinamicheskikh sistem proizvolnogo nechetnogo poryadka s dissipatsiei. III. Sistemy sedmogo poryadka”, Materialy 6 Mezhdunarodnoi konferentsii «Dinamicheskie sistemy i kompyuternye nauki: teoriya i prilozheniya» (DYSC 2024). Irkutsk, 16–20 sentyabrya 2024 g. Chast 1, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 238, VINITI, M., 2025, 69–100
M. V. Shamolin, “Invarianty odnorodnykh dinamicheskikh sistem proizvolnogo nechetnogo poryadka s dissipatsiei. IV. Sistemy devyatogo poryadka”, Materialy 6 Mezhdunarodnoi konferentsii «Dinamicheskie sistemy i kompyuternye nauki: teoriya i prilozheniya» (DYSC 2024). Irkutsk, 16–20 sentyabrya 2024 g. Chast 2, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 239, VINITI, M., 2025, 62–97
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M. V. Shamolin, “Invariants of systems having a small number of degrees of freedom with dissipation”, Moscow University Mathematics Bulletin, 79:2 (2024), 71–84
M. V Shamolin, “INVARIANTS OF GEODESIC, POTENTIAL AND DISSIPATIVE SYSTEMS WITH THREE DEGREES OF FREEDOM”, Differencialʹnye uravneniâ, 60:3 (2024), 322
José F. Cariñena, “A Geometric Approach to the Sundman Transformation and Its Applications to Integrability”, Symmetry, 16:5 (2024), 568
M. V. Shamolin, “Invariants of Seventh-Order Homogeneous Dynamical Systems with Dissipation”, Dokl. Math., 109:2 (2024), 152
M. V. Shamolin, “Invariants of Geodesic, Potential, and Dissipative Systems with
Three Degrees of Freedom”, Diff Equat, 60:3 (2024), 296
Valery V. Kozlov, “Solvable Algebras and Integrable Systems”, Regul. Chaotic Dyn., 29:5 (2024), 717–727
M. V. Shamolin, “New Cases of Integrable Ninth-Order Conservative and Dissipative Dynamical Systems”, Dokl. Math., 2024
M. V. Shamolin, “Invariants of seventh-order homogeneous dynamical systems with dissipation”, Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, 516 (2024), 65
M. V. Shamolin, “New cases of integrable ninth-order conservative and dissipative dynamical systems”, Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, 518:1 (2024), 51
M. V. Shamolin, “Invarianty odnorodnykh dinamicheskikh sistem proizvolnogo nechetnogo poryadka s dissipatsiei. I. Sistemy tretego poryadka”, Materialy Voronezhskoi mezhdunarodnoi vesennei matematicheskoi shkoly «Sovremennye metody kraevykh zadach. Pontryaginskie chteniya—XXXV», Voronezh, 26-30 aprelya 2024 g. Chast 2, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 236, VINITI RAN, M., 2024, 72–88
M. V. Shamolin, “Invarianty odnorodnykh dinamicheskikh sistem proizvolnogo nechetnogo poryadka s dissipatsiei. II. Sistemy pyatogo poryadka”, Materialy Voronezhskoi mezhdunarodnoi vesennei matematicheskoi shkoly «Sovremennye metody kraevykh zadach. Pontryaginskie chteniya—XXXV», Voronezh, 26-30 aprelya 2024 g. Chast 3, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 237, VINITI RAN, M., 2024, 49–75
S. Motonaga, K. Yagasaki, “Obstructions to integrability of nearly integrable dynamical systems near regular level sets”, Arch. Rational. Mech. Anal., 247:3 (2023), 44
M. V. Shamolin, “Invariant volume forms of geodesic, potential, and dissipative systems on a tangent bundle of a four-dimensional manifold”, Dokl. Math., 107:1 (2023), 57–63
M. V. Shamolin, “Invariant forms of geodesic, potential, and dissipative systems on tangent bundles of finite-dimensional manifolds”, Dokl. Math., 108:1 (2023), 248–255
M. V. Shamolin, “Tenzornye invarianty geodezicheskikh, potentsialnykh i dissipativnykh sistem. I. Sistemy na kasatelnykh rassloeniyakh dvumernykh mnogoobrazii”, Materialy Voronezhskoi mezhdunarodnoi zimnei matematicheskoi shkoly «Sovremennye metody teorii funktsii i smezhnye problemy», Voronezh, 27 yanvarya — 1 fevralya 2023 g. Chast 1, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 227, VINITI RAN, M., 2023, 100–128
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