Understanding d'Alembert's principle: System of Pendulums

Abstract

Lagrangian mechanics uses d'Alembert's principle of zero virtual work as an important starting point. The orthogonality of the force of constraint and virtual displacement is emphasized in literature, without a clear warning that this is true usually for a single particle system. For a system of particles connected by constraints, it is shown, that the virtual work of the entire system is zero, even though the virtual displacements of the particles are not perpendicular to the respective constraint forces. It is also demonstrated why d'Alembert's principle involves virtual work rather than the work done by constraint forces on allowed displacements.