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Abstract.A nonholonomic system, for short "NH,'' consists of a configuration space Q n , a Lagrangian L(q,q, t), a nonintegrable constraint distribution H ⊂ T Q, with dynamics governed by Lagrange-d'Alembert's principle. We present here two studies, both using adapted moving frames. In the first we explore the affine connection viewpoint. For natural Lagrangians L = T −V , where we take V = 0 for simplicity, NH-trajectories are geodesics of a (nonmetric) connection ∇ NH which mimics Levi-Civita's. Local geometric invariants are obtained by Cartan's method of equivalence.As an example, we analyze Engel's (2-4) distribution. This is the first such study for a distribution that is not strongly nonholonomic. In the second part we study * The authors thank the Brazilian funding agencies CNPq and

Key words: point vortices, Riemann surfaces, Green functions.AMS MSC(2000) ; 76B47, 31C12, 37J05 .

Can any secrets still be shed by that much studied, uniquely integrable, Elliptic Billiard? Starting by examining the family of 3-periodic trajectories and the loci of their Triangular Centers, one obtains a beautiful and variegated gallery of curves: ellipses, quartics, sextics, circles, and even a stationary point. Secondly, one notices this family conserves an intriguing ratio: Inradius-to-Circumradius. In turn this implies three conservation corollaries: (i) the sum of bounce angle cosines, (ii) the product of excentral cosines, and (iii) the ratio of excentral-to-orbit areas. Monge's Orthoptic Circle's close relation to 4-periodic Billiard trajectories is wellknown. Its geometry provided clues with which to generalize 3-periodic invariants to trajectories of an arbitrary number of edges. This was quite unexpected. Indeed, the Elliptic Billiard did surprise us!

``Rubber'' coated rolling bodies satisfy a no-twist in addition to the no slip satisfied by ``marble'' coated bodies. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at the corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2-3-5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptionalgroup G_2. The 2-3-5 nonholonomic geometries are classified in a companion paper via Cartan's equivalence method. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system with SO(3) symmetry group, total space Q = SO(3) X S^2 that can be reduced to an almost Hamiltonian system in T^*S^2 with a non-closed 2-form \omega_{NH}. In this paper we present some basic results on this reduction and as an example we discuss the sphere-sphere problem. In this example the 2-form is conformally symplectic so the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure. Using sphero-conical coordinates we verify the results by Borisov and Mamaev that the system is integrable for a ball over a plane and a rubber ball with twice the radius of a fixed internal ball.Comment: 22 pages; submitted to Regular and Chaotic Dynamic

This is an attempt to study mathematically billiards with moving baundaries. We assume that the boundary remains closed, regular and strictly mnvex. deforming periodically in time. in the normal direction. We describe the associated billiard diffenmorphism and the corresponding invariant measure. We discuss the stability of %periodic orbits and investigate the boundedness of the velocity in some precise examples. Finally, we present the Hamiltonian formalism and the symplectic structure, considering that a moving billiard is a billiard with rigid boundary on an augmented configuafion space, with a singular metric.

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