Abstract:
An extension $\Omega(G)$ is constructed for a Lie algebra $G$, and an algorithm is proposed which converts functions in involution on $G^*$ into functions in involution on $\Omega(G)^*$. Operators of “rigid body” type are constructed for $\Omega(G)$ in the case of a semisimple Lie algebra $G$; complete integrability is proved for the Euler equations on $\Omega(G)^*$ with these operators.
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\Bibitem{Tro83}
\by V.~V.~Trofimov
\paper Extensions of Lie algebras and Hamiltonian systems
\jour Math. USSR-Izv.
\yr 1984
\vol 23
\issue 3
\pages 561--578
\mathnet{http://mi.mathnet.ru/eng/im1465}
\crossref{https://doi.org/10.1070/IM1984v023n03ABEH001786}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=727757}
\zmath{https://zbmath.org/?q=an:0578.58023|0547.58024}
Linking options:
https://www.mathnet.ru/eng/im1465
https://doi.org/10.1070/IM1984v023n03ABEH001786
https://www.mathnet.ru/eng/im/v47/i6/p1303
This publication is cited in the following 11 articles:
D. V. Berzin, “O tenzornom rasshirenii odnoi klassicheskoi gamiltonovoi sistemy”, Mezhdunar. nauch.-issled. zhurn., 2014, no. 4(23), 8–9
D. V. Berzin, “O bifurkatsiyakh v tenzornom rasshirenii klassicheskoi zadachi Eilera”, Mezhdunar. nauch.-issled. zhurn., 2014, no. 7(26), 5–6
D. V. Berzin, “Osobennosti “tsentr” i “sedlo” v tenzornykh rasshireniyakh nekotorykh gamiltonovykh sistem”, Mezhdunar. nauch.-issled. zhurn., 2013, no. 2(9), 4–4
D. V. Berzin, “Perestroiki “tsentr” i “sedlo” v tenzornom rasshirenii zadachi Eilera”, Mezhdunar. nauch.-issled. zhurn., 2013, no. 3(10), 19–20
V. V. Trofimov, M. V. Shamolin, “Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems”, J. Math. Sci., 180:4 (2012), 365–530
D. V. Georgievskii, M. V. Shamolin, “Valerii Vladimirovich Trofimov”, Journal of Mathematical Sciences, 154:4 (2008), 449–461
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T. L. Mordasheva, “Canonical coordinates on orbits of a co-adjoint representation of certain semidirect products of Lie groups”, Russian Math. Surveys, 50:6 (1995), 1282–1283
V. V. Trofimov, “Canonical coordinates on orbits of the coadjoint representation of tensorial extensions of Lie groups”, Russian Math. Surveys, 49:1 (1994), 251–253
V. V. Trofimov, A. T. Fomenko, “Liouville integrability of Hamiltonian systems on Lie algebras”, Russian Math. Surveys, 39:2 (1984), 1–67
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