Geometry of nonholonomic constraints

Cushman, Richard; Kemppainen, D.; Śniatycki, Jędrzej; Bates, Larry

doi:10.1016/0034-4877(96)83625-1

Cited by 50 publications

(58 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…15). Moreover, using (15), the following property is easily proved. Note, in passing, that a similar property also applies to the gyroscopic 1-form α X .…”

Section: Koiller's Questionmentioning

confidence: 99%

On the geometry of generalized Chaplygin systems

Cantrijn

Cortés

León

et al. 2002

Math. Proc. Camb. Phil. Soc.

150

View full text Add to dashboard Cite

Some aspects of the geometry and the dynamics of generalized Chaplygin systems are investigated. First, two different but complementary approaches to the construction of the reduced dynamics are reviewed: a symplectic approach and an approach based on the theory of affine connections. Both are mutually compared and further completed. Next, a necessary and sufficient condition is derived for the existence of an invariant measure for the reduced dynamics of generalized Chaplygin systems of mechanical type. A simple example is then constructed of a generalized Chaplygin system which does not verify this condition, thereby answering in the negative a question raised by Koiller.

show abstract

“…15). Moreover, using (15), the following property is easily proved. Note, in passing, that a similar property also applies to the gyroscopic 1-form α X .…”

Section: Koiller's Questionmentioning

confidence: 99%

On the geometry of generalized Chaplygin systems

Cantrijn

Cortés

León

et al. 2002

Math. Proc. Camb. Phil. Soc.

150

View full text Add to dashboard Cite

show abstract

“…This action is not free, since S 1 leaves invariant pairs of points of the form ((0, 0, ±1), (0, 0, ω 3 )). The corresponding set of invariants is again (19) with the relations (20). Thus, the reduced orbit space M is the semialgebraic variety of R 5 described in the previous example of the rolling disk, with the same notations.…”

Section: Routh's Spherementioning

confidence: 99%

“…This problem has been treated as well, e.g., in [4,7,19,33,35]. Consider a homogeneous disk, which rolls without slipping on a horizontal plane under the influence of a vertical gravitational field of strenght g. The resulting non-holonomic system has two evident symmetry groups.…”

Section: The Rolling Diskmentioning

confidence: 99%

“…Although now the S 1 action is free, we will use again invariant theory in order to perform the reduction. Consider the invariants similar (but not equal) to (19):…”

Section: Ball Rolling On the Interior Of A Cylindermentioning

confidence: 99%

“…In recent years there has been an increasing interest in the geometric treatment of nonholonomic mechanical systems, see, e.g., [2,3,5,6,10,12,19,28,29,31,[38][39][40]. In particular, it has been recognised that the Hamiltonian formulation of such systems can be stated in terms of an almost-Poisson bracket, that is, a biderivation of functions of phase space, antisymmetric in its arguments but which does not necessarily fulfil the Jacobi identity (see, e.g., [1,11,36]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Poisson structures for reduced non-holonomic systems

Ramos¹

2004

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

Abstract. Borisov, Mamaev and Kilin have recently found certain Poisson structures with respect to which the reduced and rescaled systems of certain non-holonomic problems, involving rolling bodies without slipping, become Hamiltonian, the Hamiltonian function being the reduced energy. We study further the algebraic origin of these Poisson structures, showing that they are of rank two and therefore the mentioned rescaling is not necessary. We show that they are determined, up to a non-vanishing factor function, by the existence of a system of first-order differential equations providing two integrals of motion. We generalize the form of that Poisson structures and extend their domain of definition. We apply the theory to the rolling disk, the Routh's sphere, the ball rolling on a surface of revolution, and its special case of a ball rolling inside a cylinder.

show abstract

Dirac Brackets in Constrained Dynamics

et al. 1999

View full text Add to dashboard Cite

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.

show abstract

Geometry of nonholonomic constraints

Cited by 50 publications

References 1 publication

On the geometry of generalized Chaplygin systems

On the geometry of generalized Chaplygin systems

Poisson structures for reduced non-holonomic systems

Dirac Brackets in Constrained Dynamics

Contact Info

Product

Resources

About