Virtual work

Virtual work is the work forces would do during an imagined infinitesimal displacement compatible with a system’s constraints. At equilibrium, the total virtual work vanishes for every such displacement.

Virtual displacement: An imagined infinitesimal change in a system’s configuration that respects its constraints. Virtual work is evaluated over precisely these constraint-compatible displacements.
Mechanical work: The energy transferred by a force acting through a displacement, equal to the force’s component along that displacement times its magnitude. Virtual work uses the same force–displacement product for an imagined change.
Statics: The branch of mechanics concerned with forces on bodies at rest or in equilibrium. Virtual work provides an alternative formulation of static equilibrium.
Johann Bernoulli: A Swiss mathematician who contributed to calculus, mechanics, and the early development of the calculus of variations. His work helped establish virtual displacements as a tool for mechanical equilibrium.
Newton's laws of motion: Three foundational laws relating motion to forces, inertia, and action–reaction. They state force balance directly, while virtual work tests forces through admissible displacements.
Principle of virtual work: The equilibrium principle stating that total virtual work vanishes for every admissible virtual displacement. It applies the virtual-work quantity as a test for equilibrium.
Holonomic constraint: A constraint expressible as an equation among a system’s coordinates and possibly time. Such constraints define which infinitesimal configuration changes count as admissible.
Lagrangian mechanics: A formulation of mechanics based on a system’s kinetic and potential energies and generalized coordinates. Its equations of motion can be derived using virtual work and D'Alembert's principle.
Jean le Rond d'Alembert: An eighteenth-century French mathematician and physicist known for work in mechanics and mathematics. His principle recast dynamics in a form using virtual work.
Principle of stationary action: The principle that physical trajectories make the action stationary under suitable variations. It concerns variations of entire trajectories rather than instantaneous virtual displacements.

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Castigliano's methodRelated: It supplies an alternative energy route to structural displacements.