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The Euler-Jacobi-Lie integrability theorem. (English) Zbl 1283.34035
This article deals with a class of problems associated with the conditions for exact integrability of ordinary differential equation systems expressed in terms of the properties of tensor invariants. There is proved a general theorem on the integrability of a system of \(n\) differential equations, which admits \(n-2\) independent symmetry fields and an invariant volume \(n\)-form. The obtained results are applied to some problems of magneto-hydrodynamics.

MSC:
34C14 Symmetries, invariants of ordinary differential equations
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[1] Arnold, V. I., Kozlov, V.V., and Neîshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 1993, pp. 1–291.
[2] Olver, P. J., Applications of Lie Groups to Differential Equations, 2nd ed., Grad. Texts in Math., vol. 107, New York: Springer, 1993. · Zbl 0785.58003
[3] Kozlov, V.V., Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3), vol. 31, Berlin: Springer, 1996.
[4] Kozlov, V.V., Remarks on a Lie Theorem on the Exact Integrability of Differential Equations, Differ. Uravn., 2005, vol. 41, no. 4, pp. 553–555, 576 [Differ. Equ., 2005, vol. 41, no. 4, pp. 588–590].
[5] Kozlov, V.V., On the Theory of Integration of the Equations of Nonholonomic Mechanics, Adv. in Mech., 1985, vol. 8, no. 3, pp. 85–107 (Russian).
[6] Bolotin, S. V. and Kozlov, V. V., Symmetry Fields of Geodesic Flows, Russ. J. Math. Phys., 1995, vol. 3, no. 3, pp. 279–295. · Zbl 0912.58027
[7] Kozlov, V.V., Symmetries and Regular Behavior of Hamiltonian Systems, Chaos, 1996, vol. 6, no. 1, pp. 1–5. · Zbl 1055.37571
[8] Pesin, Ya.B., Characteristic Lyapunov Exponents and Smooth Ergodic Theory, Uspekhi Mat. Nauk, 1977, vol. 32, no. 4(196), pp. 55–112 [Russian Math. Surveys, 1977, vol. 32, no. 4, pp. 55–114].
[9] Sedov, L. I., A Course in Continuum Mechanics: Basic Equations and Analytical Techniques, Groningen: Wolters-Noordhoff, 1971. · Zbl 0308.73005
[10] Kozlov, V.V., Notes on Steady Vortex Motions of Continuous Medium, Prikl. Mat. Mekh., 1983, vol. 47, no. 2, pp. 341–342 [J. Appl. Math. Mech., 1983, vol. 47, no. 2, pp. 288–289].
[11] Arnol’d, V. I., On the Topology of Three-Dimensional Steady Flows of an Ideal Fluid, Prikl. Mat. Mekh., 1966, vol. 30, no. 1, pp. 183–185 [J. Appl. Math. Mech., 1966, vol. 30, no. 1, pp. 223–226].
[12] Borisov, A.V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443–490. · Zbl 1229.70038
[13] Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 443–464. · Zbl 1309.37049
[14] Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 170–190. · Zbl 1253.37063
[15] Kim, B., Routh Symmetry in the Chaplygin’s Rolling Ball, Regul. Chaotic Dyn., 2011, vol. 16, no. 6, pp. 663–670. · Zbl 1253.37065
[16] Kozlov, V.V., On Invariant Manifolds of Nonholonomic Systems, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 131–141. · Zbl 1323.70074
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