The central problem of constrained motion is now stated as follows:
2.
And by ( 2 ) applying the fundamental equation of constrained motion to this auxiliary unconstrained system so that the auxiliary constrained equations of motion are explicitly given by
3.
The ? "'r " "'k " are " virtual displacements ", by definition they are infinitesimal changes in the configuration of the system consistent with the constraint forces acting on the system " at an instant of time ", i . e . in such a way that the constraint forces maintain the constrained motion.
4.
Suppose we have a bead sliding around on a wire, or a swinging simple pendulum, etc . If one tracks each of the massive objects ( bead, pendulum bob, etc . ) as a particle, calculation of the motion of the particle using Newtonian mechanics would require solving for the time-varying constraint force required to keep the particle in the constrained motion ( reaction force exerted by the wire on the bead, or tension in the pendulum rod ).
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