d'Alembert's principle

Remarks

By Newton's second law, $m_i a_i = F_i + R_i$, where $R_i$ is the constraint force on particle $i$. So the sum equals $-\sum_i R_i \cdot \delta r_i$, which is zero exactly because the constraints are ideal. The gain is that the unknown constraint forces disappear from the equations. For a holonomic system with generalised coordinates $q^1, \dots, q^n$, the virtual displacements are $\delta r_i = \sum_k \frac{\partial r_i}{\partial q^k} \delta q^k$ with arbitrary $\delta q^k$. Inserting them gives Lagrange's equations $\frac{d}{dt}\frac{\partial T}{\partial \dot q^k} - \frac{\partial T}{\partial q^k} = Q_k$, with $T$ the kinetic energy and $Q_k = \sum_i F_i \cdot \frac{\partial r_i}{\partial q^k}$ the generalised forces. If the applied forces come from a potential $V$, these are the Euler–Lagrange equations of $L = T - V$. Example: take a bead of mass $m$ on a fixed smooth horizontal wire along the $x$-axis, under gravity. The virtual displacements are $(\delta x, 0, 0)$, and the principle gives $m\ddot x = 0$. The vertical reaction force never enters. Without the ideal hypothesis the principle fails: sliding friction does work on virtual displacements along the wire.

Going deeper

For a student

D'Alembert's principle is the bridge from Newton's laws to Lagrangian mechanics. It extends the principle of virtual work from statics to dynamics by treating $-m_i a_i$ as an extra force, the inertial force. Integrating it in time over variations with fixed endpoints gives Hamilton's principle for holonomic systems with a potential. It also holds for nonholonomic constraints linear in the velocities, where it leads to the equations with multipliers.