Nonholonomic and constrained variational mechanics

Lewis Andrew D.

doi:10.3934/jgm.2020013

Article Contents

2020, Volume 12, Issue 2: 165-308. Doi: 10.3934/jgm.2020013

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Nonholonomic and constrained variational mechanics

Lewis Andrew D.

Department of Mathematics and Statistics, Queeen's University, Kingston, ON K7L 3N6, Canada

Received: November 30, 2018

Revised: December 31, 2019

Early access: June 2020

Published: June 2020

Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada

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Abstract

Equations governing mechanical systems with nonholonomic constraints can be developed in two ways: (1) using the physical principles of Newtonian mechanics; (2) using a constrained variational principle. Generally, the two sets of resulting equations are not equivalent. While mechanics arises from the first of these methods, sub-Riemannian geometry is a special case of the second. Thus both sets of equations are of independent interest.

The equations in both cases are carefully derived using a novel Sobolev analysis where infinite-dimensional Hilbert manifolds are replaced with infinite-dimensional Hilbert spaces for the purposes of analysis. A useful representation of these equations is given using the so-called constrained connection derived from the system's Riemannian metric, and the constraint distribution and its orthogonal complement. In the special case of sub-Riemannian geometry, some observations are made about the affine connection formulation of the equations for extremals.

Using the affine connection formulation of the equations, the physical and variational equations are compared and conditions are given that characterise when all physical solutions arise as extremals in the variational formulation. The characterisation is complete in the real analytic case, while in the smooth case a locally constant rank assumption must be made. The main construction is that of the largest affine subbundle variety of a subbundle that is invariant under the flow of an affine vector field on the total space of a vector bundle.

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Mathematics Subject Classification: Primary: 70F25; Secondary: 34C45, 46E35, 49K05, 53B05, 53C05, 53C17, 58A30, 70G45.

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Figure 1. A depiction of $ \nu\hat{{\sigma}} $ and $ \delta\hat{{\sigma}} $. Note that $ \nu\hat{{\sigma}}_s $ is the tangent vector field for $ \hat{{\sigma}}_s $ and $ \delta\hat{{\sigma}}^t $ is the tangent vector field for $ \hat{{\sigma}}^t $

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