Constraint forces in Lagrangian mechanics

Question

Some Background

I was reading up on some elementary Lagrangian mechanics from David Morin's Classical Mechanics while also going through Goldstein's Classical Mechanics for further clarification. There was one recurring theme: in none of the worked-out problems was it checked whether the following constraints were met:

$$\sum^{N}_{i=1} \vec{f_{i}}\cdot\frac{\partial \vec{r_{i}}}{\partial q_{j}}=0\tag{1}$$

where $j=1,2,...3N-k$ ($N$ particles and $k$ holonomic constraints), $\vec {f_{i}}$ are the constraint forces, $\vec {r_{i}}$ are the position vectors of the $N$ particles, $q_{j}$ are the generalized co-ordinates.

My Question

Seeing this happening repeatedly, I have begun to wonder whether I should check whether $(1)$ is satisfied or not in the problems that I am solving.

1. Typically, my problems involve only two constraint forces: the Tension force and the Normal force. So should I check whether $(1)$ before applying E-L? If the answer is no, is it somehow mathematically guaranteed that $(1)$ is true when the constraint forces are the Normal force and the Tension force? (Supply the mathematical details, please.)

2. Also, could you please provide some general scenarios where there are chances of $(1)$ being violated? (Ignore the force of friction.)

Answer 1

• OP's eq. (1) is essentially the principle of virtual work (PVW) (if we generalize it from static to dynamical systems).

• PVW is not automatically satisfied.

• PVW is essentially equivalent to d'Alembert's principle (DAP), cf. e.g. this Phys.SE post.

• In turn DAP is essentially equivalent to Lagrange equations, cf. e.g. this Phys.SE post.

• For failure of DAP, see e.g. this Phys.SE post.

Answer 2

In a more down-to-earth argument than Qmechanic's answer, I would like to point out a specific case. Let us consider a particle rolling down a perfect hemisphere. Because the hemisphere has a fixed radius R, we can use radial coördinates $\theta$ and maybe $\phi$ for solving a problem. If you do this, then it should be easy to see that the particle cannot leave the surface of the hemisphere, i.e. the constraints cannot be violated at all!

But this can become unphysical. In the case considered, the constraint force is the normal reaction force, and this force cannot go negative. Yet, for the particle rolling down, there will come a point where this force decreases from positive, through zero, into negative. This is because our convenient coördinates assumed too much of the constraint force. It will also be the reason why basically all the relevant textbooks (I'm sure Goldstein covered this question, and IIRC Morin too) would set this question as a problem to solve. The solution is that you are supposed to manually check the constraint force for zeroes, and know for yourself when your solution needs to change from one analytic form to another, because in this case the particle falls off the hemisphere.