Hamel’s Formalism for Infinite-Dimensional Mechanical Systems

Shi, Dean; Berchenko-Kogan, Yakov; Zenkov, Dmitry V.; Bloch, Anthony M.

doi:10.1007/s00332-016-9332-7

Cited by 18 publications

(6 citation statements)

References 35 publications

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“…A direct attempt to use (3.5) and (3.13) to derive Fokker-Planck equations with standard methods gives highly non-intuitive and cumbersome formulas. It is possible that a more elegant and geometric approach developed by the method of Hamel applied to nonholonomic constraints [21,22], can be useful in approaching this problem. In this method, the velocity coordinates are chosen in such a way that the nonholonomic constraints take a particularly simple form, which may possibly lead to tractable expressions for the diffusion operator in Fokker-Planck equations.…”

Section: Discussionmentioning

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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Section: Discussionmentioning

confidence: 99%

Geometric Analysis of Noisy Perturbations to Nonholonomic Constraints

Gay–Balmaz

Putkaradze

2017

Stochastic Geometric Mechanics

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“…Below, the functional-analytic details are mostly omitted. These details and can be found in [44]. It is safe to assume that all infinite-dimensional configuration spaces are Banach manifolds, however, the results remain correct for much more general settings, such as convenient spaces.…”

Section: Infinite-dimensional Systemsmentioning

confidence: 97%

“…Being a survey, this paper leaves many technical details out. Interested readers are referred to [2,44] for such details, applications to systems with symmetry, etc. We concentrate on the formulation of the Hamilton and Lagrange-d'Alembert variational principles for Hamel's equations.…”

Section: Introductionmentioning

confidence: 99%

On Hamel’s equations

Zenkov

2016

Theor appl mech (Belgr)

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“…A direct attempt to use (3.13) gives highly nonintuitive and cumbersome formulas. It is possible that the method of Hamel applied to nonholonomic constraints (Zenkov et al 2012;Shi et al 2015) can be useful in approaching this problem. In this method, the velocity coordinates are chosen in such a way that the nonholonomic constraints take a particularly simple form, which may possibly lead to tractable expressions for the diffusion operator in Fokker-Planck equations.…”

Section: Conclusion and Further Directionsmentioning

confidence: 99%

On Noisy Extensions of Nonholonomic Constraints

Gay–Balmaz¹,

Putkaradze

2016

J Nonlinear Sci

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We propose several stochastic extensions of nonholonomic constraints for mechanical systems and study the effects on the dynamics and on the conservation laws. Our approach relies on a stochastic extension of the Lagrange-d'Alembert framework. The mechanical system we focus on is the example of a Routh sphere, i.e., a rolling unbalanced ball on the plane. We interpret the noise in the constraint as either a stochastic motion of the plane, random slip or roughness of the surface. Without the noise, this system possesses three integrals of motion: energy, Jellet and Routh. Depending on the nature of noise in the constraint, we show that either energy, or Jellet, or both integrals can be conserved, with probability 1. We also present some exact solutions for particular types of motion in terms of stochastic integrals. Next, for an arbitrary nonholonomic system, we consider two different ways of including stochasticity in the constraints. We show that when the noise preserves the linearity of the constraints, then energy is preserved. For other types of noise in the constraint, e.g., in the case of an affine noise, the energy is not conserved. We study in detail a class of Lagrangian mechanical systems on semidirect products of Lie groups, with "rolling ball type" constraints. We conclude with numerical simulations illustrating our theories, and some pedagogical examples of noise in constraints for other nonholonomic systems popular in the literature, such as the nonholonomic particle, the rolling disk and the Chaplygin sleigh.Communicated by

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Hamel’s Formalism for Infinite-Dimensional Mechanical Systems

Cited by 18 publications

References 35 publications

Geometric Analysis of Noisy Perturbations to Nonholonomic Constraints

Geometric Analysis of Noisy Perturbations to Nonholonomic Constraints

On Hamel’s equations

On Noisy Extensions of Nonholonomic Constraints

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