Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization

Ehlers, Kurt; Koiller, Jair; Montgomery, Richard; Rios, Pedro M.

doi:10.1007/0-8176-4419-9_4

Cited by 78 publications

(222 citation statements)

References 91 publications

(137 reference statements)

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“…The identity Ω nh = Ω M − Ξ appears for the first time, albeit not completely explicitly, in [4]. A proof using moving frames is given in [15] where it is also called the " J, K -formula". A different proof following the above outline is contained in [25, Prop 2.2].…”

Section: Proposition 22 (Non-holonomic Reduction)mentioning

confidence: 99%

“…This is an important question since it says something about the possible existence of asymptotic equilibria and also plays a prominent role in the theory of integration of non-holonomic systems. Correspondingly, this topic is touched upon in all of [3,6,10,15,25,22,30]. In [8], a necessary and sufficient condition for the existence of a preserved measure in terms of local coordinates on the base manifold M is given.…”

Section: E the Preserved Measurementioning

confidence: 99%

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Geometry of non-holonomic diffusion

Hochgerner¹,

Raţiu²

2015

J. Eur. Math. Soc.

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We study stochastically perturbed non-holonomic systems from a geometric point of view. In this setting, it turns out that the probabilistic properties of the perturbed system are intimately linked to the geometry of the constraint distribution. For GChaplygin systems, this yields a stochastic criterion for the existence of a smooth preserved measure. As an application of our results we consider the motion planning problem for the noisy two-wheeled robot and the noisy snakeboard.

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Section: Proposition 22 (Non-holonomic Reduction)mentioning

confidence: 99%

Section: E the Preserved Measurementioning

confidence: 99%

Geometry of non-holonomic diffusion

Hochgerner¹,

Raţiu²

2015

J. Eur. Math. Soc.

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“…III A is carried over to the Hamiltonian side. Specifically, we define the horizontal lift hl P q : T * qQ → P q by (see Ehlers et al 53 )…”

Section: A the Geometry Of Weakly Degenerate Chaplygin Systemsmentioning

confidence: 99%

Hamilton–Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints

Leok

Ohsawa

Sosa

2012

Journal of Mathematical Physics

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We extend Hamilton-Jacobi theory to Lagrange-Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton-Jacobi equation as the Dirac-Hamilton-Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton-Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmetries, which we refer to as weakly degenerate Chaplygin systems, that arise as simplified models of nonholonomic mechanical systems; these systems are shown to reduce to non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian systems defined with non-closed two-forms. Accordingly, the Dirac-Hamilton-Jacobi equation reduces to a variant of the nonholonomic Hamilton-Jacobi equation associated with the reduced system. We illustrate through a few examples how the Dirac-Hamilton-Jacobi equation can be used to exactly integrate the equations of motion. C 2012 American Institute of Physics. [http://dx

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“…Moreover, as we shall see in the context of nonholonomic mechanics in Section 3.2, the Weitzenböck torsion will also be responsible for the "pseudogyroscopic" constraint reaction force (see footnote 11 of [18], pg. 106 of [15] and [31]) present in the reduced equations of motion.…”

Section: 10)mentioning

confidence: 98%

“…Our goal in this paper is twofold: to show that the Weitzenböck connection is responsible for the fictitious pseudogyroscopic force (see footnote 11 of [18], pg. 106 of [15] and [31]) appearing in the non-coordinate basis treatment of mechanics, and to apply the insights gained from this to the study of Chaplygin Hamiltonization [22,24].…”

Section: Introductionmentioning

confidence: 99%

The Weitzenböck connection and time reparameterization in nonholonomic mechanics

Fernandez¹,

Bloch²

2011

Journal of Mathematical Physics

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We show that the torsion of the Weitzenböck connection is responsible for the fictitious pseudogyroscopic force experienced by a general mechanical system in a non-coordinate moving frame. In particular, we show that for the class of mechanical systems subjected to non-integrable constraints known as nonabelian nonholonomic Chaplygin systems, the constraint reaction force directly depends on this torsion. For these Chaplygin systems, we show how this torsional force can in some cases be removed by an appropriate choice of frame depending on a multiplier f (q), linking these results to the process of Chaplygin Hamiltonization through time reparameterization. Lastly, we show that the cyclic symmetries of f in some cases lead to the existence of momentum conservation laws for the original nonholonomic system, and illustrate the results through several examples.

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Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization

Cited by 78 publications

References 91 publications

Geometry of non-holonomic diffusion

Geometry of non-holonomic diffusion

Hamilton–Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints

The Weitzenböck connection and time reparameterization in nonholonomic mechanics

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