A virtual displacement of a system refers to a change in the configuration of the system as the result of any arbitrary infinitesimal change of the coordinates δri, consistent with the forces and constraints imposed on the system at the given instant t. (-Goldstein)

The principle of virtual work states that

If the net virtual work (during any virtual displacement δr) done by all external forces on a rigid body consistent with the constraints imposed on the body, is zero then the body is in equilibrium.

However, this principle requires that internal frictional forces do no work during the virtual displacement (all other types of internal forces cancel each other).

                                                       δW= ∑ F. δri = 0                                        (summed over all particles)

The principle above considers only static constraints on the system, when the constraint forces do no work at all and only hold the system in place. However, in the dynamics problems, we can extend the principle above to include the forces of constraints as well, by expressing them as inertial forces.

In this case, the net work on the system changes to the sum of the work done by the external(or applied) forces and work done by inertial forces(describing constraints):

                                                       δW= ∑ F. δri +∑ Fi*. δri = 0                    (summed over all particles)

Or, equivalently,


The equation above summarizes d’Alembert’s principle which states that

the sum of the differences between the forces acting on a system of mass particles and the time derivatives of the momenta of the system itself along any virtual displacement consistent with the constraints of the system, is zero. (-Wikipedia)

This principle is somewhat similar to the Newton’s second law of motion. However, it is not that easy to prove that they are equivalent. Anyhow, this is not very useful in the present form. We need to derive equations of motion from it in order to apply it in real life situations.

more on virtual work

D’Alembert’s Principle
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