Paula Balseiro

This paper studies hamiltonization of nonholonomic systems using geometric tools. By making use of symmetries and suitable first integrals of the system, we explicitly define a global 2-form for which the gauge transformed nonholonomic bracket gives rise to a new bracket on the reduced space codifying the nonholonomic dynamics and carrying an almost symplectic foliation (determined by the common level sets of the first integrals). In appropriate coordinates, this 2-form is shown to agree with the one previously introduced locally in [34]. We use our coordinate-free viewpoint to study various geometric features of the reduced brackets. We apply our formulas to obtain a new geometric proof of the hamiltonization of a homogeneous ball rolling without sliding in the interior side of a convex surface of revolution using our formulas.

In this paper we study the relation between conserved quantities of nonholonomic systems and the hamiltonization problem employing the geometric methods of [1,3]. We illustrate the theory with classical examples describing the dynamics of solids of revolution rolling without sliding on a plane. In these cases, using the existence of two conserved quantities we obtain, by means of 'gauge transformations' and symmetry reduction, genuine Poisson brackets describing the reduced dynamics.

We study the existence of first integrals in nonholonomic systems with symmetry. First we define the concept of M-cotangent lift of a vector field on a manifold Q in order to unify the works [Balseiro P.,

In this paper we study the reduction of a nonholonomic system by a group of symmetries in two steps. Using the so-called 'vertical-symmetry' condition, we first perform a 'compression' of the nonholonomic system leading to an almost symplectic manifold. Second, we perform an (almost) symplectic reduction, relying on the existence of a momentum map. In this case, we verify that the resulting (almost) symplectic reduced spaces are the leaves of the characteristic foliation of the reduced nonholonomic bracket. On each leaf we study the (Lagrangian) equations of motion, obtaining a nonholonomic version of the Lagrange-Routh equations, and we analyze the existence of a conformal factor for the reduced bracket in terms of the existence of conformal factors for the almost symplectic leaves. We also relate the conditions for the existence of a momentum map for the compressed system with gauge transformations by 2-forms. The results are illustrated in three different examples.

In this paper we study the Jacobiator (the cyclic sum that vanishes when the Jacobi identity holds) of the almost Poisson brackets describing nonholonomic systems. We revisit the local formula for the Jacobiator established by Koon and Marsden in \cite{MarsdenKoon} using suitable local coordinates and explain how it is related to the global formula obtained in \cite{paula}, based on the choice of a complement to the constraint distribution. We use an example to illustrate the benefits of the coordinate-free viewpoint.