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Liouville integrability of Hamiltonian systems on Lie algebras. (English. Russian original) Zbl 0549.58024
Russ. Math. Surv. 39, No. 2, 1-67 (1984); translation from Usp. Mat. Nauk 39, No. 2(236), 3-56 (1984).
This is a review of various aspects of integrability in the Liouville sense. The Liouville theorem is given both in its well-known classical formulation and in its noncommutative version. Some examples of integrability are discussed, such as the motion of rigid body, the dynamics of Toda lattice, etc. Methods of constructing involutive sets of functions are described. Some results on integrability of geodesic flows on two-dimensional surfaces are presented. Sufficient conditions for integrability of geodesic flows on coadjoint bundles of Lie groups are given.
Reviewer: I.Ya.Dorfman

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
17B99 Lie algebras and Lie superalgebras
53C35 Differential geometry of symmetric spaces
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