D'Alembert's principle

D'Alembert's principle states that applied forces and reversed inertial forces do zero total virtual work for virtual displacements compatible with a system's constraints. It recasts dynamics as a condition resembling static equilibrium.

Virtual work: The work a force would do during an infinitesimal displacement allowed by a system's constraints. The principle applies virtual work to applied and inertial forces together.
Newton's laws of motion: Three classical laws relating motion to inertia, forces, and action-reaction pairs. The principle rewrites Newton's second law as a virtual-work balance.
Jean le Rond d'Alembert: An eighteenth-century French mathematician and physicist who formulated the principle bearing his name. His 1743 treatise presented the principle as a foundation for treating dynamics through statics.
Lagrange's equations: Differential equations for mechanical motion derived from a system's Lagrangian and generalized coordinates. They follow by applying the principle to generalized coordinates and ideal constraints.
Newtonian mechanics: Classical mechanics that describes motion using forces, masses, and Newton's laws. It writes force balances directly, while D'Alembert's principle recasts them as virtual-work balances.
Virtual displacement: An infinitesimal change in configuration consistent with a system's constraints at a fixed instant. These allowed variations are the displacements over which the total virtual work vanishes.
Generalized coordinates: Independent variables that specify a mechanical system's configuration, including its constraints. They describe admissible configurations without requiring Cartesian coordinates for every particle.
Traité de dynamique: Jean le Rond d'Alembert's 1743 treatise on the foundations of dynamics. It gave the principle its original systematic presentation.
Constrained mechanical system: A mechanical system whose possible configurations or motions are restricted by specified conditions. The principle handles such systems without explicitly solving for every constraint force.
Lagrangian mechanics: A formulation of classical mechanics based on generalized coordinates and the difference between kinetic and potential energy. It is the resulting equation-based framework, rather than the principle's virtual-work statement.

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