d’Alembert–Lagrange analytical dynamics for nonholonomic systems

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.