Criterion for the Validity of D'Alembert's Equations of Motion

For mechanical systems subject to time dependent, holonomic constraints, the principle of virtual work is apparently required to derive D'Alembert's equations of motion. This is in contrast to situations where the constraints are time independent, where the equations of motion can be derived by standard arguments using vector calculus and linear algebra. Attempts to apply this method when some of the constraints have an explicit time dependence lead extra terms in the equations of motion. These apparent terms are removed by appealing to the principle of virtual work. In this work we show that, for the cases of universal, revolute, and telescopic joints between two rigid bodies of which one's motion is specified externally, these terms apparently vanish identically when the computer algebra system Mathematica is used. This leads us to provide lengthy but elementary analytic proofs that the extra terms vanish identically for the three cases which, we believe, are exhaustive for real mechanical system.