Dirac Brackets in Constrained Dynamics

Abstract

Non-holonomic constraints, both in the Lagragian and Hamiltonian formalism, are discussed from the geometrical viewpoint of implicit differential equations. A precise statement of both problems is presented remarking the similarities and differences with other classical problems with constraints. In our discussion, apart from a constraint submanifold, a field of permitted directions and a system of reaction forces are given, the later being in principle unrelated to the constraint submanifold. An implicit differential equation is associated to a non-holonomic problem using the Tulczyjew's geometrical description of the Legendre transformation. The integrable part of this implicit differential equation is extracted using an adapted version of the integrability algorithm. Moreover, sufficient conditions are found that guarantees the compatibility of the non-holonomic problem, i.e., that assures that the integrability algorithm stops at first step, and moreover it implies the existence of a vector field whose integral curves are the solutions to the problem. In addition this vector field turns out to be a second order differential equation. These compatibility conditions are shown to include as particular cases many others obtained previously by other authors. Several examples and further lines of development of the subject are also discussed.

Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems

“…In particular, we construct the nonholonomic bracket of [30,25,21] using Corollary 3.4 and the framework for nonholonomic mechanics described in Bates and Sniatycki [2]. In Proposition 4.3 we show that the dynamics associated with this bracket coincide with the formulation of nonholonomic mechanics on almost Dirac structures considered in [32,33,23]. Next, we define the notion of dynamical gauge transformations for a nonholonomic system, and define a family F of almost Poisson brackets, possessing the same characteristic distribution, and that describe our nonholonomic system.…”

“…These are more general geometric objects that provide the framework in which gauge transformations are more natural. These structures had been already considered in connection to nonholonomic mechanics by Yoshimura and Marsden [32,33], and by Jotz and Ratiu [23]. However, the issue of Hamiltonization and the incorporation of gauge transformations are not treated in these works.…”

“…Instead, they are written with respect to an almost Poisson bracket that fails to satisfy the Jacobi identity. This formulation has its origins in [34,29,23] and others.…”

“…Let us consider a nonholonomic system on a constraint phase space M, equipped with the almost Poisson bracket {·, ·} nh [34,29,23], known as the nonholonomic bracket, and Hamiltonian function H M . The (almost) Hamiltonian vector field X nh , defined by {·, ·} nh and H M , governs the dynamics of the system.…”

Nonholonomic constraints: A new viewpoint

“…In particular, if the original structure was a Lie algebroid, then the new algebroid bracket is automatically skew-symmetric, so we deal with a quasi-Lie algebroid. One can the associate with the sequence of procedures, like reduction by symmetries and passing to a nonholonomic constraint, the sequence of the corresponding novel structures serving as appropriate geometrical tools in describing the systems: All this is of course closely related to the discovery of the role of the nonholonomic quasi-Poisson brackets [22,35,15,4], this time not in the Hamilton but in the Lagrange picture.…”