On nonholonomic systems and variational principles

Abstract

A generalization of Chetaev's principle for a class of higher order nonholonomic constraintsWe consider the compatibility of the equations of motion which follow from d'Alembert's principle in the case of a general autonomous nonholonomic mechanical system in N dimensions with those equations which follow for the same system by assuming the validity of a specific variational action principle, in which the nonholonomic conditions are implemented by means of the multiplication rule in the calculus of variations. The equations of motion which follow from the principle of d'Alembert are not identical to the equations which follow from the variational action principle. We give a proof that the solutions to the equations of motion which follow from d'Alembert's principle do not in general satisfy the equations which follow from the action principle with nonholonomic constraints. Thus the principle of d'Alembert and the minimal action principle involving the multiplication rule are not compatible in the case of systems with nonholonomic constraints. For simplicity the proof is given for autonomous systems only, with one general nonholonomic constraint, which is linear in the generalized velocities of the system.

Atomistic Hydrodynamics and the Dynamical Hydrophobic Effect in Porous Graphene

“…where u(r) is the streaming velocity of the fluid evaluated at point r, and dots denote time derivatives. This constraint is nonholonomic and cannot be treated easily using Euler-Lagrange or Hamiltonian dynamics [45,46], so we turn instead to Gauss's principle of least constraint [24,47,48]. The Gaussian cost function C is…”

Fractional Dynamics of Relativistic Particle

“…Hamiltonain form [36,37,38,39,40]. Note that nonholonomic variational equations must be used since the fractional equations which follow from the d'Alembert-Lagrange principle (and the Hamilton's principle) do not in general equivalent the equations which follow from the action principle with nonholonomic constraints [23,24,25,29,30].…”

On the compatibility of non-holonomic systems and certain related variational systems

“…The questions discussed by Flannery are by no means new; they have been discussed in the literature in several papers. For a selection of references other than those given by Flannery, I refer to some of the references contained in a recent paper 2 by Cronström and Raita [3].…”

“…As such, this validates the assertion that the solutions to the two different types of equations of motion are different in general, at least in a space of three dimensions. It has only recently been proved, [3], [5], that this is also the case generally in configuration spaces of dimension N ≥ 3. Below I discuss an improved version of the proofs in [3] and [5], separately for N = 3 and N ≥ 4.…”

“…It has only recently been proved, [3], [5], that this is also the case generally in configuration spaces of dimension N ≥ 3. Below I discuss an improved version of the proofs in [3] and [5], separately for N = 3 and N ≥ 4.…”