Examples of gauge conservation laws in nonholonomic systems

Bates, Larry; Graumann, Hugo; MacDonell, Creighton

doi:10.1016/0034-4877(96)84069-9

Cited by 41 publications

(84 citation statements)

References 2 publications

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“…In contrast with the general case, this case is completely and explicitly solvable, as it is well known, see, e.g., [4,8,32,33]. However, we will give an independent treatment.…”

Section: Ball Rolling On the Interior Of A Cylindermentioning

confidence: 96%

“…For example, to what extent the mentioned Poisson structures can be useful to investigate the intimate nature of these and maybe other non-holonomic systems, for example in order to characterize their integrability properties [3,21,22]; see also the recent work [23]. Another question could be to clarify the origin of the system of differential equations giving the conservation laws for the mentioned reduced non-holonomic systems, see also [4,10,19,32,40] and references therein.…”

Section: Discussionmentioning

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%

“…This problem has been treated as well, e.g., in [4,7,20,33,35]. Consider a sphere of mass m and radius r with its center of mass at a distance α (0 < α < r) from its geometric center.…”

Section: Routh's Spherementioning

confidence: 99%

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Poisson structures for reduced non-holonomic systems

Ramos¹

2004

J. Phys. A: Math. Gen.

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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.

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“…In contrast with the general case, this case is completely and explicitly solvable, as it is well known, see, e.g., [4,8,32,33]. However, we will give an independent treatment.…”

Section: Ball Rolling On the Interior Of A Cylindermentioning

confidence: 96%

Section: Discussionmentioning

confidence: 99%

Section: The Rolling Diskmentioning

confidence: 99%

“…This problem has been treated as well, e.g., in [4,7,20,33,35]. Consider a sphere of mass m and radius r with its center of mass at a distance α (0 < α < r) from its geometric center.…”

Section: Routh's Spherementioning

confidence: 99%

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Poisson structures for reduced non-holonomic systems

Ramos¹

2004

J. Phys. A: Math. Gen.

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“…Here the constraint distribution does have length 1 everywhere, however, after performing the appropriate computations, we obtain β = − y 1 + y 2 dy, which is clearly exact: β = df , with f = 1 2 ln 1 1 + y 2 . Then, k = 1 1 + y 2 and the invariant measure defined by k ω n leads, using Euler's integrating factor technique, to the constant of the motion ϕ =ẋ 1 + y 2 which was also obtained by different methods in [4,7,13].…”

Section: A Counter Examplementioning

confidence: 99%

On the geometry of generalized Chaplygin systems

Cantrijn

Cortés

León

et al. 2002

Math. Proc. Camb. Phil. Soc.

150

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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.

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