d'Alembert's principle
/ dal-om-BAIR /
A profound reframing of Newton's second law that quietly removes the constraint forces from the picture. The idea is to think of the term -m a as an inertial force, so that at every instant the applied forces plus this inertial force are in balance. Then the constraint forces, which do no work under the allowed (virtual) displacements, drop out entirely.
The statement is: the sum over all particles of (F_i - m_i a_i) dot delta r_i = 0 for every virtual displacement delta r_i consistent with the (workless) constraints. Because ideal constraint forces are perpendicular to the allowed motion, they contribute nothing to this sum. Rewriting this in generalized coordinates is precisely what produces the Euler-Lagrange equations, so d'Alembert's principle is the bridge that carries you from Newton to Lagrange.
It is the true logical foundation of Lagrangian mechanics for constrained systems, and it is the correct starting point for nonholonomic systems, where blindly varying the action gives wrong answers. A virtual displacement delta r is an imagined, instantaneous displacement consistent with the constraints at a frozen instant of time; it is not an actual displacement occurring over a time interval dt.
Setting the accelerations to zero recovers the principle of virtual work of statics: the applied forces alone do no net virtual work in equilibrium.
Statics is the a = 0 special case.
It assumes constraint forces do no virtual work (ideal constraints); sliding kinetic friction violates this and must be reintroduced as an applied generalized force.