What was the professor talking about regarding constraint forces? I am taking a mechanics class at university. Last week we started describing movement over a curve; we are given a natural parametrisation $s(t)$, and then we have the following relations:
$$ \vec{r}(t) = \vec{r}(s(t)) $$
$$ \vec{v} = \dot{\vec{r}} = \frac{d \vec{r}}{ds} \dot{s} = \vec{e}_t \dot{s} $$
$$ \vec{a} = \ddot{s} \vec{e}_t + \dot{s} \frac{d \vec{e}_t}{ds} \dot{s} = \ddot{s} \vec{e}_t + \kappa \dot{s}^2 \vec{e}_n $$
We said the coordinate system $\{ \vec{e}_t, \vec{e}_n, \vec{e}_b \}$ is called an intrinsic coordinate system (it consists of unit vector in tangential, normal and binormal direction). This far, everything's ok.
Then he wrote the Newton's equation:
$$ \vec{F}' = m \vec{a} = m (\ddot{s} \vec{e}_t + \kappa \dot{s}^2 \vec{e}_n ) $$
He stated that something "feels wrong" about this, since the number of equations and unknowns doesn't match. This is where I first got confused: what are those equations and unknowns that he mentions? If we write Newton's formula in vector form, taking the intrinsic basis, we get:
$$ (m \ddot{s}, \kappa \dot{s}^2, 0) = (F_x', F_y', F_z') $$
I see we get three equations, but I'm not sure what the unknowns are, and not even which quantities we are supposed to know; do we know $F$ or do we know $s, \dot{s}, \ddot{s}$? Again, what exactly are the unknowns? (I hope I'm right at least in counting the three equations)
Then he said that in fact $\vec{F}' = \vec{F} + \vec{S} $, where $\vec{F}$ is called active force and $\vec{S}$ is called, I believe, constraint force (in my language it's something like force of bonds; if my translation is wrong, let me know, I'll change the question). He further said that all forces which have a physical origin and act on particles are active forces, while constraint forces pull material points toward curves.
He than said that we are still in troubles. If we write:
$$ m (\ddot{s} \vec{e}_t + \kappa \dot{s}^2 \vec{e}_n ) = \vec{F} + \vec{S}$$
$$ m \ddot{s} = \vec{F} \vec{e}_t + \vec{S} \vec{e}_t $$
$$ m \kappa \dot{s}^2 = \vec{F} \vec{e}_n + \vec{S} \vec{e}_n $$
$$ 0 = \vec{F} \vec{e}_b + \vec{S} \vec{e}_b $$
Now, he said, there are too many unknowns. Once again, I have no idea what he is talking about here.
He added that we can fix this if we add a constitutive equation, namely:
$$ \vec{S} \cdot \vec{e}_t = 0 $$
He said that if this is true, then we are talking about smooth curves (in the sense of no friction).
So, all in all, my question can be summarized as follows:
What are the equations and unknowns that my professor is talking about and that we are counting? How many are there? What are the equations and unknowns in the second part, when we add $\vec{S}$? What exactly are the equations and unknowns when we add the constitutive equation?
 A: For a constraint force, think of a rollercoaster. If you want model such a system you could specify the location of the track by some function $f(x)$ or by some more elaborate parametrized curve as you mentioned in your question.
Ignoring any structural concerns, you would want the car of the rollercoaster to remain stuck to the track at all times. This is exactly what constraint forces will do for you. They make sure that given a certain initial velocity, force of gravity, etc... the car will remain on the trajectory you want it to follow.
There are more simple examples such as a bead on a circular loop. In order to satisfy the constraint that the bead should remain attached to the loop, you need to introduce constraints that make sure your problem is posed in a physically sensible way.
In the constitutive equation we have the statement that ${S}\cdot e_t=0$. If you recall the definition of the dot product, this mean that the constraint forces are orthogonal to the direction along the curve. In the rollercoaster example, this would mean that the forces from the car pushing on the track due to acceleration or gravity would be cancelled by the normal force due to the structural integrity of the track. As long as these forces do not have a component along the direction of movement (which would imply that $S\cdot e_t\not=0$) they do not contribute to slowing down or accelerating the "particle" along the direction of the curve (which is what $e_t$ - the tangential direction - is representing).
This is a bit of a qualitative answer but I hope it helps with your intuition. Perhaps somebody else can clarify in a more technical way.
