Starting from the general kinetic principle of d’Alembert/Lagrange, an energetic proof of the sufficiency conditions for equilibrium (known as Principle of Virtual Work) is presented. It is clearly demonstrated why to maintain equilibrium requires that, in addition to the familiar vanishing of the virtual work of the impressed forces on the originally motionless system, its geometrical (holonomic) constraints be explicitly time independent (stationary) and its nonintegrable kinematical (nonholonomic) ones be linear and homogeneous in the generalized velocities (catastatic).
On the Sufficiency of the Principle of Virtual Work for Mechanical Equilibrium: A Critical Reexamination
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Papastavridis, J. G. (September 1, 1989). "On the Sufficiency of the Principle of Virtual Work for Mechanical Equilibrium: A Critical Reexamination." ASME. J. Appl. Mech. September 1989; 56(3): 704–707. https://doi.org/10.1115/1.3176150
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