The Wagner Curvature Tensor in Nonholonomic Mechanics

We present the classical Wagner construction from 1935 of the curvature tensor for completely nonholonomic manifolds in both invariant and coordinate way. The starting point is the Shouten curvature tensor for nonholonomic connection introduced by Vranceanu and Shouten. We illustrate the construction on two mechanical examples: the case of a homogeneous disc rolling without sliding on a horizontal plane and the case of a homogeneous ball rolling without sliding on a fixed sphere. In the second case we study the conditions on the ratio of diameters of the ball and the sphere to obtain a flat space - with the Wagner curvature tensor equal zero.Comment: 22 page

Bifurcations of Liouville Tori in Elliptical Billiards

A detailed description of topology of integrable billiard systems is given. For elliptical billiards and geodesic billiards on ellipsoid, the corresponding Fomenko graphs are constructed.Comment: 18 pages, 15 figure

Systems of the Kowalevski type and discriminantly separable polynomials

Starting from the notion of discriminantly separable polynomials of degree two in each of three variables, we construct a class of integrable dynamical systems. These systems can be integrated explicitly in genus two theta-functions in a procedure which is similar to the classical one for the Kowalevski top. The discriminnatly separable polynomials play the role of the Kowalevski fundamental equation. The natural examples include the Sokolov systems and the Jurdjevic elasticae.Comment: 29 pages, 0 figures. arXiv admin note: substantial text overlap with arXiv:1106.577