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Several problems on dynamical systems and mechanics. (English) Zbl 1160.37026
The author discusses five open problems and, connected with them, the theory of dynamic systems of classical and quantum mechanics. These are the integrability of geodesic flow on algebraic surfaces; integrability and polynomial integrals for Hamiltonian systems with the Hamiltonian \(H=\frac{1}{2}(a y^{2}_{1}+ 2 b y_1 y_2+cy^{2}_{2}) + V(x_1, x_2)\), where \(x_1, x_2\) mod \(2\pi\) are angular coordinates and \(y_1, y_2\) the conjugate momenta; some problems concerned with polynomial conservation laws in quantum mechanics; two problems on instability for autonomous system of ODEs \(\dot x =v(x), x\in \mathbb{R}^n\) with invariant measure with smooth density, when \(\text{div}(\rho v)=0, \rho(x)<0\); problem of rolling motion of a disc.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37J30 Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria)
70K50 Bifurcations and instability for nonlinear problems in mechanics
81Q50 Quantum chaos
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