Quantum tops as examples of commuting differential operators

We study the quantum analogs of tops on Lie algebras $so(4)$ and $e(3)$ represented by differential operators.Comment: 24 p

Non-homogeneous systems of hydrodynamic type possessing Lax representations

We consider 1+1 - dimensional non-homogeneous systems of hydrodynamic type that possess Lax representations with movable singularities. We present a construction, which provides a wide class of examples of such systems with arbitrary number of components. In the two-component case a classification is given.Comment: 22 pages, latex, minor change

Atlas of two-dimensional irreversible conservative lagrangian mechanical systems with a second quadratic integral

This paper aims at the most comprehensive and systematic construction and tabulation of mechanical systems that admit a second invariant, quadratic in velocities, other than the Hamiltonian. The configuration space is in general a 2D Riemannian or pseudo-Riemannian manifold and the determination of its geometry is a part of the process of solution. Forces acting on the system include a part derived from a scalar potential and a part derived from a vector potential, associated with terms linear in velocities in the Lagrangian function of the system. The last cause time-irreversibility of the system. We construct 41 multi-parameter integrable systems of the type described in the title mostly on Riemannian manifolds. They are mostly new and cover all previously known systems as special cases, corresponding to special values of the parameters. Those include all known cases of motion of a particle in the plane and all known cases in the dynamics of rigid body. In the last field we introduce a new integrable case related to Steklov's case of motion of a body in a liquid. Several new cases of motion in the plane, on the sphere and on the pseudo-sphere or in the hyperbolic plane are found as special cases. Prospective applications in mathematics and physics are also pointed out.Comment: Paper to be published in "Journal of Mathematical Physics", Vol. 48, issue 7, July 200

Chaplygin ball over a fixed sphere: explicit integration

We consider a nonholonomic system describing a rolling of a dynamically non-symmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel--Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconical) coordinates on the Poisson sphere, which can be useful in other integrable problems. Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.Comment: This is an extended version of the paper to be published in Regular and Chaotic Dynamics, Vol. 13 (2008), No. 6. Contains 20 pages and 2 figure