Selection of virtual displacements for general nonholonomic constraints

Determine the appropriate definition of virtual displacements δr for mechanical systems subject to general nonholonomic constraints Vv(r, ṙ, t) = 0 (v = 1, …, K), particularly for nonlinear constraints, such that the selection is compatible with the constraint functions and enables the d’Alembert principle to yield the correct equations of motion.

Background

The paper studies discrete mechanical systems with velocity-dependent (nonholonomic) constraints and derives equations of motion algebraically via the d’Alembert principle. A central issue in such systems is how to define virtual displacements so that the virtual work of constraint forces vanishes and the resulting equations are correct.

The authors introduce two general classes of virtual displacements, labeled (A) and (B), and later link them to Četaev and vakonomic conditions, respectively. While they develop equivalence results under specific assumptions (notably A = C and particular assignments in terms of constraint gradients), they explicitly note that, in the general (especially nonlinear) case, selecting the appropriate virtual displacements remains an open question.