Hamiltonization of Solids of Revolution Through Reduction

Abstract

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.

Gauge Momenta as Casimir Functions of Nonholonomic Systems

“…Indeed, if r ≤ 2 then the section Λ of ∧ 3 (D * ) of Lemma 5.4 vanishes and the bracket Π Λ nh in Theorem 5.6 coincides with Π nh . An instance of this general situation is encountered by Balseiro [6] in her treatment of the nonholonomic particle.…”

Conserved quantities and Hamiltonization of nonholonomic systems

“…Following our construction, we revisit three examples that were known to be hamiltonizable via gauge transformations by 2-forms. We show how the previous diagram explains the hamiltonization procedure in each situation, explaining why we get a twisted Poisson bracket in the cases of the Chaplygin ball [33,5,1] and the snakeboard [1,4], and a Poisson bracket in the case of the solids of revolution rolling on a plane [2,34]. We then present a new example of hamiltonization via gauge transformations: a homogeneous ball rolling without sliding on a convex surface of revolution [16,27,39,53,54,60], showing that the dynamics is described by the Poisson bracket {•, •} 1 red .…”

“…The hamiltonization problem studies whether a nonholonomic system admits a hamiltonian formulation after a reduction by symmetries, and much work has been done on this problem in recent years, see e.g. [2,5,13,14,15,26,30,34,42,43,47,59]. The possibility of writing the reduced equations of motion in hamiltonian form is central in the study of various aspects of nonholonomic systems, such as integrability, Hamilton-Jacobi theory and even numerical methods (e.g.…”

“…But even when {•, •} red is not Poisson, it has been observed in many examples that there could be other brackets on M/G relating X red and H red as in (1.1), and which are Poisson (or close to being Poisson, e.g. conformally Poisson, or twisted Poisson [44,55]), see [1,2,5,13,14,33,34,53]. The issue is then finding systematic ways to produce such brackets on M/G.…”

“…In the context of hamiltonization, one is interested in dynamical gauge transformations, which have the additional property that the modified bracket {•, •} B still describes the nonholonomic dynamics, in the sense that X nh = {•, H M } B , see [5]. In [1,2,4,32,33], and specially in [34], it has been observed that the existence of such a 2-form B producing Poisson-type brackets on M/G is related to the presence of conserved quantities of the system called horizontal gauge momenta [7,28,29].…”