×

zbMATH — the first resource for mathematics

Multi-Hamiltonian property of a linear system with quadratic invariant. (English. Russian original) Zbl 07090628
St. Petersbg. Math. J. 30, No. 5, 877-883 (2019); translation from Algebra Anal. 30, No. 5, 159-168 (2018).
Summary: It is shown that a nondegenerate linear system that admits a nondegenerate quadratic from as a first integral can be represented in several different ways as a Hamiltonian system of differential equations; we present a “complete” family of the corresponding symplectic structures and Hamiltonians. We discuss possible generalizations of this result including the case of linear systems of differential equations with periodic coefficients.

MSC:
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 1 V. V. Kozlov, Linear systems with a quadratic integral, Prikl. Mat. Mekh. 56 (1992), no. 6, 900-906; English transl., Appl. Math. Mech.  56 (1992), no. 6, 803-809.
[2] 2 D. V. Treshch\"ev and A. A. Shkalikov, On the Hamiltonian property of linear dynamical systems in Hilbert space, Mat. Zametki  101 (2017), no. 6, 911-918; English transl., Math. Notes 101 (2017), no. 5-6, 1033-1039. · Zbl 06769031
[3] 3 V. Volterra, Sopra una classe di equazioni dinamiche, Atti Accad. Sci. Torino 33 (1897-98), 451-475.
[4] 4 E. B. Gledzer, F. V. Dolzhanski\i , and A. M. Obukhov, Systems of hydrodynamic type and their application, Nauka, Moscow, 1981. (Russian) · Zbl 0512.76003
[5] 5 I. A. Bizyaev and V. V. Kozlov, Homogeneous systems with quadratic integrals, Lie-Poisson quasi-brackets and the Kovalevskaya method, Mat. Sb.  206 (2015), no. 12, 29-54; English transl., Sb. Math. 206 (2015), no. 11-12, 1682-1706. · Zbl 1358.37099
[6] 6 V. V. Kozlov, Linear Hamiltonian systems: quadratic integrals, singular subspaces and stability, Regul. Chaotic Dyn.  23 (2018), no. 1, 26-46. · Zbl 1400.37061
[7] 7 J. Williamson, An algebraic problem involving the involutory integrals of linear dynamical systems, Amer. J. Math.  62 (1940), 881-911. · JFM 66.0991.02
[8] 8 F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), no. 5, 1156-1162. · Zbl 0383.35065
[9] 9 P. J. Olver, Applications of Lie groups to differential equations, Grad. Texts in Math., vol. 107, Springer-Verlag, New York, 1993. · Zbl 0785.58003
[10] 10 A. V. Borisov and I. S. Manaev, Modern methods in the theory of integrable systems, Inst. Komp’yut. Issled., Izhevsk, 2003. (Russian)
[11] 11 A. Wintner, On the linear conservative dynamical systems, Ann. Mat. Pura Appl. 13 (1934), no. 1, 105-112. · Zbl 0009.37904
[12] 12 H. Kocak, Linear Hamiltonian systems are integrable with quadratics, J. Math. Phys. 23 (1982), no. 12, 2375-2380. · Zbl 0507.70015
[13] 13 V. V. Kozlov, On the mechanism of stability loss, Differ. Uravn. 45 (2009), no. 4, 496-505; English transl., Differ. Equ.  45 (2009), no. 4, 510-519. · Zbl 1375.70018
[14] 14 H. Weyl, The classical groups. Their envariants and representations, Princeton Univ. Press, Princeton, NJ, 1939.
[15] 15 V. V. Kozlov, The Liouville property of invariant measures of completely integrable systems and the Monge-Amp\`ere, Mat. Zametki  53 (1993), no. 4, 45-52; English transl., Math. Notes 53 (1993), no. 3-4, 389-393.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.