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Poisson structures and Lie algebras in Hamiltonian mechanics. (Пуассоновы структуры и алгебры Ли в гамильтоновой механике.) (Russian) Zbl 1010.70002
Izhevsk: Izd. dom “Udmurtskiĭ Univ.”, Nauchno-Izdatel’skiĭ Tsentr “Regulyarnaya i Khaoticheskaya Dinamika”. 464 p. (1999).
The book contains a survey of Hamiltonian mechanics and its connections with the theory of Lie algebras and integrability of dynamical systems.
The contents of the book are as follows: 1. Poisson brackets and Hamiltonian formalism; 2. Poisson brackets in rigid body dynamics; 3. Hamiltonian formalism in celestial mechanics; 4. Hamiltonian dynamics of vortex structures; 5. Many-particle systems. In eight appendices the authors describe some applications of Hamiltonian formalism to nonholonomic systems, to singular orbits of co-adjoint representations of group \(\mathrm{SO}(n)\) and \(E(n)\), to nonintegrability of Dyson system, to three-body problem etc.
Chapter 1 is devoted to a general description of Hamiltonian formalism and to the integrability problems for Hamiltonian systems and for their reductions. Chapter 2 contains some applications of Kovalevskaya-Lyapunov method to the integrability analysis of the motion of a rigid body including the case of Dirac mechanics. The authors consider geodesic flows on spheres \(S^2\) and \(S^3\), and give a multidimensional generalization of the analysis of gyroscope motion. In chapter 3, the dynamics is studied in three-dimensional spaces of constant curvature. In particular, the authors consider non-Euclidean versions of Kepler problem and restricted three-body problem. A list of all known integrable cases in the spaces of constant curvature related to celestial mechanics is given.
Chapter 4 describes the dynamics of vortex systems on plane, sphere, and inside a circular domain. The main attention is focused on the motion of three-vortex and \(N\)-vortex systems. The authors also present some applications to the analysis of Lotka-Volterra system. The final chapter 5 contains an analysis of Toda lattices, of their integrability, and describes the corresponding Lax representation.

70-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems
70Hxx Hamiltonian and Lagrangian mechanics
74G65 Energy minimization in equilibrium problems in solid mechanics
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70E40 Integrable cases of motion in rigid body dynamics