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d’Alembert–Lagrange analytical dynamics for nonholonomic systems

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10.1063/1.3559128

### Abstract

The d’Alembert–Lagrange principle (DLP) is designed primarily for dynamical systems under ideal geometric constraints. Although it can also cover linear-velocity constraints, its application to nonlinear kinematic constraints has so far remained elusive, mainly because there is no clear method whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint. On recognition that the commutation rule traditionally accepted for velocity displacements in Lagrangian dynamics implies displaced states that do not satisfy the kinematic constraints, we show how the property of possible displaced states can be utilized *ab initio* so as to provide an appropriate set of linear auxiliary conditions on the displacements, which can be adjoined via Lagrange's multipliers to the d’Alembert–Lagrange equation to yield the equations of state, and also new transpositional relations for nonholonomic systems. The equations of state so obtained for systems under general nonlinear velocity and acceleration constraints are shown to be identical with those derived (in Appendix A) from the quite different Gauss principle. The present advance therefore solves a long outstanding problem on the application of DLP to ideal nonholonomic systems and, as an aside, provides validity to axioms as the Chetaev rule, previously left theoretically unjustified. A more general transpositional form of the Boltzmann–Hamel equation is also obtained.

© 2011 American Institute of Physics

Received 17 July 2010
Accepted 02 February 2011
Published online 18 March 2011

Acknowledgments: This research has been supported by (U.S.) Air Force Office of Scientific Research (US-AFOSR) Grant No. FA95500-06-1-0212 and National Science Foundation (NSF) Grant No. 04-00438.

Article outline:

I. INTRODUCTION

II. D’ALEMBERT–LAGRANGE PRINCIPLE

A. Background: Geometric constraints embedded and adjoined

B. Background: Linear-velocity constraints adjoined and embedded

C. Transpositional rule for exactly integrable systems

III. GENERAL KINEMATIC CONSTRAINTS: NEW RESULTS

A. Equations of state for homogeneous velocity constraints

B. Equations of state for general velocity constraints

C. Equations of state for general acceleration constraints

IV. TRANSPOSITIONAL RELATIONS FOR VELOCITY CONSTRAINTS

A. Nonintegrable velocity constraints

1. Subrules

2. Integrable velocity constraints

B. Transpositional form of the d’Alembert–Lagrange principle

C. The δ(*dq*) − *d*(δ*q*)-relation

D. Higher-order transpositional relations

V. TRANSPOSITIONAL RELATIONS FOR ACCELERATION CONSTRAINTS

A. Primary transpositional rule

1. Subrules

B. Higher-order transpositional relations

VI. CONSTRAINED PRINCIPLES

A. Constrained Hamilton's Principle

B. Axiomatic constrained principles

VII. TRANSPOSITIONAL RELATIONS FOR QUASIVELOCITIES IN LINEAR-VELOCITY CONSTRAINTS

A. Transpositional form of DLP in quasivelocities

B. Boltzmann–Hamel equation for linear-velocity constraints

VIII. SUMMARY AND CONCLUSION

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2011-03-18

2014-04-21

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