When Nonholonomic Constraints are not-so

We discussed how a holonomic constraint can be written as and effectively means that you’ve chosen one generalized coordinate too many (in that you can now write one as a function of the others). But let’s say that our constraint also included the velocities: What if we had a constraint, however, that had the form Is there a condition under which such a constraint, which looks like it’s nonholonomic, is actually holonomic? Well, if we can find a function $g(q_i, t)$ such that $a_i = \frac{\partial g}{\partial q_i}$ and $b=\frac{\partial g}{\partial t},$ then we may rewrite our constraint

But this is actually just

which can readily be integrated to

which is a holonomic constraint.