Stochastic chaplygin systems

Hochgerner, Simon

doi:10.1016/s0034-4877(10)80010-2

Cited by 6 publications

(6 citation statements)

References 20 publications

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“…Collectively, rolling particles have different phase behaviours than those that slide [27]. Yet despite their intriguing dynamics, rolling has been considered in stochastic settings only for simple systems such as a rolling ball or sled [28][29][30], or as a noisy relaxation of the rolling constraint itself [31]. This paper studies a natural model of stochastic, rolling particles, with the aim of determining how rolling could affect quantities that are macroscopically measurable.…”

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confidence: 99%

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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confidence: 99%

Stochastic disks that roll

Holmes-Cerfon

2016

Phys. Rev. E

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“…These considerations were updated and recast in the language of geometric mechanics in [3], inspiring new considerations such as symmetry reduction and Noether's theorem in the presence of stochasticity. In this context, the papers [4,5] introduced mechanical systems subjected to stochastic forces while obeying non-holonomic constraints. The physical background for the systems considered there could be understood as the dynamics of a microscopic object bombarded by outside molecules, while preserving a non-holonomic constraint, such as rolling contact.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of non-holonomic systems with stochastic transport

Holm

Putkaradze

2018

Proc. R. Soc. A.

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This paper formulates a variational approach for treating observational uncertainty and/or computational model errors as stochastic transport in dynamical systems governed by action principles under non-holonomic constraints. For this purpose, we derive, analyse and numerically study the example of an unbalanced spherical ball rolling under gravity along a stochastic path. Our approach uses the Hamilton-Pontryagin variational principle, constrained by a stochastic rolling condition, which we show is equivalent to the corresponding stochastic Lagrange-d'Alembert principle. In the example of the rolling ball, the stochasticity represents uncertainty in the observation and/or error in the computational simulation of the angular velocity of rolling. The influence of the stochasticity on the deterministically conserved quantities is investigated both analytically and numerically. Our approach applies to a wide variety of stochastic, non-holonomically constrained systems, because it preserves the mathematical properties inherited from the variational principle.

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“…Stochastic perturbations of Hamiltonian systems have been considered before in [6][7][8][9][10][11]. Closer to our interests here, the stochastic extensions of nonholonomically constrained systems, obeying exact nonholonomic constraints, was studied in [12,13]. In that work, the focus was on physical systems that are sensitive to noise, while preserving the nature of the nonholonomic constraints exactly.…”

Section: Introductionmentioning

confidence: 99%

Geometric Analysis of Noisy Perturbations to Nonholonomic Constraints

Gay–Balmaz

Putkaradze

2017

Stochastic Geometric Mechanics

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We propose two types of stochastic extensions of nonholonomic constraints for mechanical systems. Our approach relies on a stochastic extension of the Lagranged'Alembert framework. We consider in details the case of invariant nonholonomic systems on the group of rotations and on the special Euclidean group. Based on this, we then develop two types of stochastic deformations of the Suslov problem and study the possibility of extending to the stochastic case the preservation of some of its integrals of motion such as the Kharlamova or Clebsch-Tisserand integrals.

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Stochastic chaplygin systems

Cited by 6 publications

References 20 publications

Stochastic disks that roll

Stochastic disks that roll

Dynamics of non-holonomic systems with stochastic transport

Geometric Analysis of Noisy Perturbations to Nonholonomic Constraints

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