Problem 6.8 on p. 39 in David Morin's The Lagrangian Method gives a stick pivoted at the origin and rotating around the pivot with constant angular velocity $\dot{\alpha}$ (which is given as $\omega$ in the document). A small mass $m$ can slide frictionless along the stick. The problems asks to find the conserved energy of the system, which it gives as $$E=\frac{1}{2}m(\dot{r}^2-r^2\dot{\alpha}^2)$$ in the solutions.
My question is not how to obtain the solution, which I understand for this problem. My questions are related to generalized coordinates and constraints used here to see if I understood the methodology for more complex systems.
At first I made the mistake to consider two generalized coordinates $q_1=r$ and $q_2=\alpha$, because it felt natural to consider all velocities ($\dot{r}$ and $\dot{\alpha}$) when dealing with energies.
It turned out that the problem has two constraints ($m=2$) and one particle ($n=1$), which leaves me with just one generalized coordinate ($S=3n-m=1$). My first constraint is $z=0$.
My second constraint should be $\alpha - \dot{\alpha}t=0$.
Is this second constraint chosen correctly and is it an example of nonholonomic constraints, because it depends on a velocity?
Do I always obtain a wrong result when choosing more generalized coordinates than I have degrees of freedom? I could think of this problem very general without constraining $\dot{\alpha}$ at all. Once I have a solution for the conserved energy, I could make different assumptions like $\dot{\alpha}=const$ or $\dot{r}=const$ to see how the conserved energy changes with different assumptions. But then I would have used more general coordinates during my problem description than I have degrees of freedom through my assumptions, which would be wrong?
