Are constraint forces infinite?

Question

A lot of authors claim that mechanical constraints are idealizations obtained by allowing enforcing forces to be infinite. But I either disagree or don't know what they mean. The only case where I would find it to be true is these forces were impulsive, i.e., the velocity would change abruptly through the means of Dirac delta impulses.

On the other hand, all the text books present a theory where the constraint forces are always bounded and smooth (either as Lagrange multipliers or the limit of a very strong potential). This makes me think they can never be infinite and there is no reason for them being infinite. The only possibility I see is that only the stiffness of the force potential goes to infinity (while also assuming that the constraint violations/oscillations are small).

Which view is right? And also, would there be any damping involved?

Answer 1

OP already seems to have thought long and hard about this and makes good points. In this answer we will review the argument for why constraint forces could be infinite.

We will assume that OP talks about holonomic$^1$ constraints. To be concrete, let the constraint be that some generalized coordinate vanishes $$q~\approx~0.$$ We can implement the constraint

1. ideally via a Lagrange multiplier term$^2$
   $$L_1~=~L_0 +\lambda q,\tag{1}$$ where the constraint force can be identified with the Lagrange multiplier $\lambda$;

2. or pragmatically via a stiff spring potential $$L_2~=~L_0 -\frac{k}{2} q^2,\tag{2}$$ where the spring constant $k$ is very large.

If $E$ denotes a characteristic energy available to the system, it is reasonable to expect $$\frac{k}{2} q^2~\lesssim~ E,$$ or $$|q|~\lesssim~{\cal O}(k^{-1/2}).$$ Hence the spring force $$ |F|~=~|-kq|~\lesssim~{\cal O}(k^{1/2})~\to~\infty\quad{\rm for}\quad k~\to~\infty.$$ In other words, the spring force $F$ could be very large, and unbounded from above as $k\to\infty$. Of course, it might not be large at all the time. E.g. there could be an oscillatory pattern.

In particular, if one identifies the spring force $F$ in model 2 with the constraint force $\lambda$ in model 1, one can argue that the (absolute value of) the latter could be very large, cf. OP's title question.

3. We can unify model 1 & 2 via the Lagrangian $$L_3~=~L_0 +\frac{\lambda^2}{2k} +\lambda q.\tag{3}$$ The EOM for $\lambda$ is $$\lambda~\approx~ -kq.$$

   • On one hand $$ \lim_{k\to\infty} L_3~=~L_1.$$

   • On the other hand, for finite $k>0$, if we integrate out/eliminate $\lambda$ from $L_3$ via its EOM, we exactly get $L_2$.

--

$^1$ Semi-holonomic constraints are quite subtle, cf. e.g. this Phys.SE post.

$^2$ The Lagrangian $L_0$ could in principle contain many degrees of freedom, i.e. depend on many generalized coordinates. We will assume that the system has at least 1 more coordinate than $q$, so that the system is non-trivial.

Answer 2

I think there is no single answer to this question; it depends on circumstances. If you have a three-dimensional region of space, with a sphere sitting in it, for example, then you might have a particle whose motion is constrained to lie on the surface of the sphere. In setting up Lagrangian mechanics for this case, you don't have to invoke any notion of force; you can deal with it as a constraint. But if you then ask what kind of force would result in the same motion, then it is a 'hard wall' type of force, which rises to infinity as soon as the particle position moves away from the sphere's surface.

One could also notice that we commonly do mechanics in a three-dimensional space, and you could say that the particles were 'constrained' to stay in the 3D space and not wander off into some 4th dimension. In this case one is concerned with a physical problem in which there simply is not any 4th spatial dimension. So we would not ordinarily call that a constraint, but mathematically speaking it has the same effect as a constraint.

Overall I think the answer is that some cases where we invoke a constraint, that is a convenient way to handle what really is the result of a force, and other cases are not.

Answer 3

In practice you're of course right, however allowing for infinite forces is the easiest way to handwave exact fulfillment of the constraints.

As argued in Qmechanics answer, in practice the energy is limited, then the stiffer you make the spring constant the less max deviation you get from the exact constraints. The idea is that you can then compute the dynamics for any finite $k$, obtain the limit $k\to\infty$, and there the deviations will tend to zero but the maximum forces still stay finite. In other words, you could replace the $\tfrac{k}2q^2$ potential with a capped version, $\max\{E, \tfrac{k}2q^2\}$ or something smooth like $E\cdot\tanh(\tfrac{k}{2E}q^2)$. That is actually not sufficient to guarantee the forces will stay bounded though.

An example where you do get infinite forces is a particle constrained to a trail $$ q\in \bigl\{(x,y) : y = \sin(\exp(x))\bigr\} $$ with no other forces. As the particle travels rightwards, it will pass every $y$-maximum with the same $x$-velocity due to conservation of kinetic energy. But there it has a $y$-acceleration $\propto -\exp(x)$, which grows without bounds.

(Real-world application: a high speed train rolling into a tight corner without braking first won't be saved by the fact that the rails are almost perfect constraints.)

To get a bound on the maximum force, what you need is a bound on the curvature of the constraint manifold, in addition to a bound on the energy.

Answer 4

When you simplify your coordinate system with mechanical constraints, you are assuming that:

1. The system will stay on the constrained path; and
2. There will be no energy stored in the constraints.

As a real mechanical constraint becomes stiffer, i.e., the magnitude of its restoring force as a function of deviation increases, it approaches this ideal more closely, in that you must apply increasing force to achieve a certain deviation or to store a certain amount of energy.

In the limit, the restoring force approaches infinity and the assumptions hold as long as the forces against the constraints are finite.

This is close to what @Qmechcanic said, except that the key idea is that no finite force could store any energy in the constraints, so there can be no oscillations against the constraints or anything like that.

Answer 5

This is physics, so the question should boil down to what you'd measure in an experiment. What would you see if you built a constrained mechanism and placed a load cell at the point where the constraining force is applied? The answers above give you theoretical approaches, noting that a load cell is actually a very stiff spring with a very sensitive strain gauge. If you're doing real physics and not textbook problems, you should build the mechanism and use it to test your results.

Or perhaps, the question should be whether the idealization will get you in trouble, leading to unrealistic results. This, of course, affects all applications of mathematical infinity in physics. In general, it's a difficult question. Textbooks tend to map the paths through the minefield that you'll need to solve their problems, but at the end you don't know where the mines actually are.