Identify appropriate virtual displacements for general nonholonomic constraints

Determine a rigorous and general prescription for the class of virtual displacements δr to be used in the d’Alembert principle for mechanical systems subject to general (including nonlinear, velocity-dependent) nonholonomic constraints Ψν(r, ṙ, t) = 0. The selection must be compatible with the constraints and be explicitly expressed in terms of the constraint functions Ψν, extending the unambiguous holonomic case to the general nonholonomic setting.

Background

The paper studies how to derive equations of motion for general nonholonomic systems via an algebraic application of the d’Alembert principle. A key step is specifying admissible virtual displacements δr for which the virtual work of reaction forces vanishes. While this specification is standard for holonomic constraints, it lacks a universally accepted form for general nonholonomic constraints.

To address this, the authors formalize two classes of virtual displacements: (A) displacements satisfying A(r, ṙ, t)δr = 0, and (B) displacements satisfying B(r, ṙ, t)δr + C(r, ṙ, t)δṙ = 0. Under a particular identification with the constraint functions—namely A = C = ∇{ṙ}Ψ and B = ∇{r}Ψ—these correspond to the Cetaev and vakonomic approaches, respectively. Despite results showing equivalence under certain assumptions, the general problem of specifying the appropriate δr for nonholonomic constraints remains explicitly open.