The Nonholonomy of the Rolling Sphere

Johnson, Brody Dylan

doi:10.1080/00029890.2007.11920439

Cited by 15 publications

(17 citation statements)

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“…While it turns out that discs in the plane are holonomic, a cluster of spheres should be nonholonomic: it can access a space that is higher-dimensional than the space along which it is constrained to move. This should be true because a single sphere rolling on a plane is non-holonomic [50,51]. Geometrically, it lives on a sub-Riemannian manifold [64][65][66], an object which has been little studied in the physics literature.…”

Section: Outlook and Conclusionmentioning

confidence: 99%

“…Such a roughness entropy would be similar in spirit to a vibrational entropy, but different in form because it should not necessarily be possible to obtain it as a harmonic expansion of a function of variables in configuration space only. Indeed, the constraint which models a sphere rolling on a plane is nonholonomic [50,51], so any jiggling about the constraint cannot depend only on the location and overall rotation of the sphere. Even for a pair of discs, one may wish to allow irreversible, nonharmonic slippage about their points of contact.…”

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confidence: 99%

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Stochastic disks that roll

Holmes-Cerfon

2016

Phys. Rev. E

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We study a model of rolling particles subject to stochastic fluctuations, which may be relevant in systems of nano-or micro-scale particles where rolling is an approximation for strong static friction. We consider the simplest possible non-trivial system: a linear polymer of three of discs constrained to remain in contact, and immersed in an equilibrium heat bath so the internal angle of the polymer changes due to stochastic fluctuations. We compare two cases: one where the discs can slide relative to each other, and the other where they are constrained to roll, like gears. Starting from the Langevin equations with arbitrary linear velocity constraints, we use formal homogenization theory to derive the overdamped equations that describe the process in configuration space only. The resulting dynamics have the formal structure of a Brownian motion on a Riemannian or subRiemannian manifold, depending on if the velocity constraints are holonomic or non-holonomic. We use this to compute the trimer's equilibrium distribution both with, and without, the rolling constraints. Surprisingly, the two distributions are different. We suggest two possible interpretations of this result: either (i) dry friction (or other dissipative, nonequilibrium forces) changes basic thermodynamic quantities like the free energy of a system, a statement that could be tested experimentally, or (ii) as a lesson in modeling rolling or friction more generally as a velocity constraint when stochastic fluctuations are present. In the latter case, we speculate there could be a "roughness" entropy whose inclusion as an effective force could compensate the constraint and preserve classical Boltzmann statistics. Regardless of the interpretation, our calculation shows the word "rolling" must be used with care when stochastic fluctuations are present.Particles that live on the nano-or microscale commonly have short-ranged interactions, so their surfaces come close enough that surface frictional effects may be important. For example, recent experiments and simulations have shown that tangential frictional forces between rough, and otherwise stochastic particles, are probably the origin of the shear-thickening behaviour of many materials [1,2]. Other studies demonstrate that sticky tethers attached to particle surfaces can change their dynamics [3,4]. Since one promising method of creating colloids with programmable interactions is to coat them with strands of DNA [5][6][7][8], which could impede their relative sliding, this could have major implications for their assembly pathways and hence structures that can be formed by selfassembly. On these scales it is extremely difficult to measure the particles' rotational degrees of freedom, so one must resort to indirect methods to determine whether tangential frictional forces are present [9,10]. Therefore, it would be highly desirable to find a simpler way to quantify these forces, via macroscopic measurements of spatial positions only.While the mascroscopic effect of dry friction has been studied in detail i...

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Section: Outlook and Conclusionmentioning

confidence: 99%

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confidence: 99%

Stochastic disks that roll

Holmes-Cerfon

2016

Phys. Rev. E

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“…The interest in this particular case can be traced back as far as the late 19th century, for instance, see [5,6]. A detailed exposition of the non-holonomy of the rolling sphere is presented in [12].…”

Section: Introductionmentioning

confidence: 99%

Geometric conditions for the existence of a rolling without twisting or slipping

Molina

Grong

2014

Communications on Pure &Amp; Applied Analysis

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We give a complete answer to the question of when two curves in two different Riemannian manifolds can be seen as trajectories of rolling one manifold on the other without twisting or slipping. We show that up to technical hypotheses, a rolling along these curves exists if and only if the geodesic curvatures of each curve coincide. By using the anti-developments of the curves, which we claim can seen of as a generalization of the geodesic curvatures, we are able to extend the result to arbitrary absolutely continuous curves. For a manifold of constant sectional curvature rolling on itself, two such curves can only differ by an isometry. In the case of surfaces, we give conditions for when loops in the manifolds lift to loops in the configuration space of the rolling.2000 Mathematics Subject Classification. 37J60, 53A55, 53A17.

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“…We see that, after traveling on a circle the sphere is rotated by 2πΩ/α with respect to an axis tilted with respect to the plane; this is the nonholonomy treated in [10] and [24].…”

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confidence: 99%

“…The problem we are considering is therefore a kinematic rather than a dynamic one: the trajectory of the contact point of the sphere and the surface is dictated externally and the rolling constraint is imposed. We make contact with recent approaches that consider the same problem [9,10] (but on a plane), and in particular, we address the nice question posed by Brockett and Dai [11]: a sphere lies on a table and is made to rotate by a flat plane on top of it, parallel to the table. The question is: if every point of the plane describes a circle, what is the trajectory and motion of the sphere?…”

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confidence: 99%