Virtual work and d’Alembert’s principle

A virtual displacement compares nearby configurations at the same instant. For holonomic constraints, differentiate the position map with respect to independent coordinates while keeping time fixed. This is different from the actual displacement during a short interval, which also contains any motion of the track or support. A bead on a moving wire provides a useful test: its virtual displacement is tangent to the wire's instantaneous shape, although its actual velocity can have a normal component imposed by the wire.

Call a constraint ideal when its reaction forces have zero total virtual work for every allowed virtual displacement. This is an assumption about the contact model, not a consequence of merely writing a holonomic equation. A smooth stationary surface satisfies it, while sliding friction along that surface generally does not. A moving ideal support can still transfer real energy because zero virtual work at fixed time does not imply zero power along the actual motion.

Separate each force into an applied contribution and an ideal reaction. Subtract mass times acceleration from applied force, dot with a virtual displacement, and sum over particles. Newton's law makes this sum vanish after the ideal reactions drop out. The minus sign on inertia matters: d’Alembert's principle is a convenient equation of dynamic balance, not a claim that an actual additional force has been discovered. Retaining this sign makes an independent force-balance check straightforward.

Generalized force is defined by the coefficient of a generalized virtual displacement in applied virtual work. For a length coordinate it has force units; for an angle it has torque units. If a potential depends only on configuration and time, its generalized force is minus its coordinate derivative. A motor torque, a dashpot, or friction not already represented by that potential stays on the right-hand side of Lagrange's equation. Counting the same force in both places doubles its effect.

For a plane pendulum, take the angle positive away from the downward vertical. Increasing that angle raises the bob locally when the angle is positive. Gravity therefore has negative generalized force, while the string tension is perpendicular to the allowed virtual displacement. The inertia coefficient is mass times length squared, so dividing a torque by it produces angular acceleration. If the string becomes slack, the original circle constraint fails and this reduced equation no longer describes the motion.

Use the vector lab as a projection check before doing the worked calculation. Set a magnitude and compare positive and negative angles to predict the sign of one Cartesian component. Its single vector can represent a chosen force direction, but it does not display a moving constraint, an acceleration balance, or a second independent vector. You must supply the tangent direction analytically. The teaching point is to check signs geometrically rather than to let an attractive diagram replace the force model.