# Prelim1 Review

# When Nonholonomic Constraints are not-so

We discussed how a *holonomic* constraint can be written as
\(f(q_i, t) = 0,\)
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:
\(f(q_i, \dot{q}_i, t) = 0.\)
What if we had a constraint, however, that had the form
\(f(\dot{q}_i, t) = a_i \dot{q}_i + b?\)
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

\[f(\dot{q}_i, t) = \frac{dg}{dt} = 0,\]which can readily be integrated to

\[g(q_i, t) - \textrm{constant} = 0,\]which is a holonomic constraint.