VI-8 CHAPTER 6. THE LAGRANGIAN METHOD
For now, let’s just recognize that in some cases S
0
is a minimum, in some cases it is a saddle
p oint, and it is never a maximum. “Least action” is therefore a misnomer.
5. It is sometimes said that nature has a “purpose,” in that it seeks to take the path that
pro duces the minimum action. In view of the second remark above, this is incorrect. In fact,
nature does exactly the opposite. It takes every path, treating them all on equal footing. We
end up seeing only the path with a stationary action, due to the way the quantum mechanical
phases add. It would be a harsh requirement, indeed, to demand that nature make a “global”
decision (that is, to compare paths that are separated by large distances), and to choose the
one with the smallest action. Instead, we see that everything takes place on a “local” scale.
Nearby phases simply add, and everything works out automatically.
When an archer shoots an arrow through the air, the aim is made possible by all the other
arrows taking all the other nearby paths, each with essentially the same action. Likewise,
when you walk down the street with a certain destination in mind, you’re not alone. . .
When walking, I know that my aim
Is caused by the ghosts with my name.
And although I can’t see
Where they walk next to me,
I know they’re all there, just the same.
6. Consider a function, f(x), of one variable (for ease of terminology). Let f(b) be a local
minimum of f. There are two basic properties of this minimum. The first is that f(b) is
smaller than all nearby values. The second is that the slope of f is zero at b. From the above
remarks, we see that (as far as the action S is concerned) the first property is completely
irrelevant, and the second one is the whole point. In other words, saddle points (and maxima,
although we showed above that these never exist for S) are just as good as minima, as far as
the constructive addition of the e
iS/¯h
phases is concerned.
7. Given that classical mechanics is an approximate theory, while quantum mechanics is the
(more) correct one, it is quite silly to justify the principle of stationary action by demon-
strating its equivalence with F = ma, as we did above. We should be doing it the other way
around. However, because our intuition is based on F = ma, it’s easier to start with F = ma
as the given fact, rather than calling upon the latent quantum-mechanical intuition hidden
deep within all of us. Maybe someday. . .
At any rate, in more advanced theories dealing with fundamental issues concerning the tiny
building blocks of matter (where actions are of the same order of magnitude as ¯h), the
approximate F = ma theory is invalid, and you have to use the Lagrangian method.
8. When dealing with a system in which a non-conservative force such as friction is present, the
Lagrangian method loses much of its appeal. The reason for this is that non-conservative
forces don’t have a potential energy associated with them, so there isn’t a specific V (x) that
you can write down in the Lagrangian. Although friction forces can in fact be incorporated in
the Lagrangian method, you have to include them in the E-L equations essentially by hand.
We won’t deal with non-conservative forces in this chapter. ♣
6.3 Forces of constraint
A nice thing about the Lagrangian method is that we are free to impose any given constraints
at the beginning of the problem, thereby immediately reducing the number of variables. This
is always done (perhaps without thinking) whenever a particle is constrained to move on a
wire or surface, etc. Often we are concerned not with the exact nature of the forces doing
the constraining, but only with the resulting motion, given that the constraints hold. By
imposing the constraints at the outset, we can find the motion, but we can’t say anything
about the constraining forces.
If we want to determine the constraining forces, we must take a different approach. The
main idea of the strategy, as we will show below, is that we must not impose the constraints