Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems

In this paper we study the problem of Hamiltonization of nonholonomic systems from a geometric point of view. We use gauge transformations by 2-forms (in the sense of Severa and Weinstein [29]) to construct different almost Poisson structures describing the same nonholonomic system. In the presence of symmetries, we observe that these almost Poisson structures, although gauge related, may have fundamentally different properties after reduction, and that brackets that Hamiltonize the problem may be found within this family. We illustrate this framework with the example of rigid bodies with generalized rolling constraints, including the Chaplygin sphere rolling problem. We also see how twisted Poisson brackets appear naturally in nonholonomic mechanics through these examples

First Integrals and symmetries of nonholonomic systems

In nonholonomic mechanics, the presence of constraints in the velocities breaks the well-under\-stood link between symmetries and first integrals of holonomic systems, expressed in Noether's Theorem. However there is a known special class of first integrals of nonholonomic systems generated by vector fields tangent to the group orbits, called {\it horizontal gauge momenta}, that suggest that some version of this link should still hold. In this paper we prove that, under certain conditions on the symmetry Lie group, the (nonholonomic) momentum map is conserved along the nonholonomic dynamics, thus extending Noether Theorem to the nonholonomic framework. Our analysis leads to a constructive method, with fundamental consequences to the integrability of some nonholonomic systems as well as their hamiltonization. We apply our results to three paradigmatic examples: the snakeboard, a solid of revolution rolling without sliding on a plane and a heavy homogeneous ball that rolls without sliding inside a convex surface of revolution. In particular, for the snakeboard we show the existence of a new horizontal gauge momentum that reveals new aspects of its integrability

A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries

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., Arch. Ration. Mech. Anal. 214 (2014), 453-501, arXiv:1301.1091], [Fosse F., Ramos A., Sansonetto N., Regul. Chaotic Dyn. 12 (2007), 579588], and [Fosse F., Giacobbe A., Sansonetto N., Rep. Math. Phys. 62 (2008), 345-367]. Second, we study gauge symmetries and gauge momenta, in the cases in which there are the symmetries that satisfy the so-called vertical symmetry condition. Under such condition we can predict the number of linearly independent first integrals (that are gauge momenta). We illustrate the theory with two examples

Orthogonal projections and the dynamics of constrained mechanical systems

A coordinate-free version of the approach to mechanical systems with non-ideal restrictions developed by Udwadia (2002) and Udwadia and Kalaba (2002) in a series of articles is introduced. Some of its properties are then reinterpreted in a general geometric setting in terms of orthogonal projections. A geometric view of other aspects of constrained systems, inspired by their insight, is also presented.Facultad de Ciencias Exacta

Orthogonal projections and the dynamics of constrained mechanical systems

A coordinate-free version of the approach to mechanical systems with non-ideal restrictions developed by Udwadia (2002) and Udwadia and Kalaba (2002) in a series of articles is introduced. Some of its properties are then reinterpreted in a general geometric setting in terms of orthogonal projections. A geometric view of other aspects of constrained systems, inspired by their insight, is also presented.Facultad de Ciencias Exacta