Transpositional rule for constrained systems

This paper investigates the dynamics of nonholonomic mechanical systems, focusing on fundamental variational assumptions and the role of the transpositional rule. We analyze how the Cetaev condition and the first variation of constraints define compatible virtual displacements for systems subject to kinematic constraints, including those nonlinear in generalized velocities. The study explores the necessary conditions for commutation relations to hold, clarifying their impact on the consistency of the derived equations of motion. By detailing the interplay between these variational identities and the Lagrangian derivatives of constraint functions, we elucidate the differences between equations of motion formulated via the d'Alembert--Lagrange principle and those obtained from extended time-integral variational principles. This work aims to provide a clearer theoretical framework for applying these core principles to nonholonomic dynamics.