Back in August this was a long thread. It might be more appropriate
to start a new long thread, but I'll stick to the old one for now.
I just reread that thread, and although it originated with a
discussion about teaching non-advanced physics, I was surprised to
not see even one reference to the principle of least action (the
Principle).
And here is where I want to jump off: I do not intend this as a
thread on what to teach in a non-advanced physics class. Or even what
to teach in any classroom at this point - for now, this is just for
me.
Aside from the question of energy being real, or not, the Principle
is relatively simple to understand as a concept, intuitive, and in
treatments involving classical mechanics, has the benefit of being
able to derive those equations of motion that are otherwise only
arrived at through Newton's F=m*dp/dt. It would seem that the
Principle does give energy a privileged position over force (but see
below).
I'm not so much geared toward the question of energy as "privileged"
as I am the quest to figure out a minimum set of underlying
principles behind physics. It seems one could do so far worse than
the Principle.
Having said that, the Principle seems to have shortcomings. Each of
these can surely be the basis for its own new thread, but I'll
summarize my questions/observations here:
1. The Principle seems most at home in classical mechanics. It has a
development in optics (Fermat's Principle?) but I haven't seen a good
treatment of manufacturing one from the other (IOW, some kind of
"optics lagrangian"). I've read that the Principle also can be
developed in thermodynamics, but I have not found such. Jackson
spends a good bit of time developing it for classical
electrodynamics. Despite all this, the Principle never takes a center
stage that I can tell, except for its role in classical mechanics.
Why does it not seem as "useful" elsewhere as it does in classical
mechanics? [1]
2. A criticism leveled at the classical mechanics version of the
Principle is that although it can used to derive Newton's Laws (or at
least the 2nd one), one needs Newton's Laws to do so. I can sort of
see this criticism, but I'm not sure I believe it - it seems more due
to the need to define force as an energy gradient, indicating, if
nothing else, that maybe energy alone can't be primary and
fundamental?
3. One can rigorously derive the differential equation for L, but
then the form for L (that is, T - V) seems to have an air of
hand-waving to it, and is possibly the origin of point 2 above (often
with the sense of "we have to write it this way to get the answer we
want"). Landau and Lifshitz goes to some effort to show why L, for a
free particle, should be proportional to v^2, as does Jackson, but I
feel insufficiently educated or convinced. The explicit form for V
seems even more problematic, requiring a definition like V =
-integral[F dot dx], which leads us to point 2 again ("we have to
write it this way to get the answer we want").
4. Foreshadowing the notion of a force being an energy gradient,
non-conservative forces introduce significant complications of
analysis. And then there's the seeming proliferation of
classifications of constraints. Goldstein's treatment seems to be an
example of a pretty complete analysis of this issue for classical
mechanics, but also suggests (my view) that the whole topic is so
complicated that it detracts from the usefulness of the Principle.
And, if these complications are present for classical mechanics,
wouldn't they be present for the other areas mentioned in #1?
If I was deconstructing my understanding of physics and rebuilding it
from the pieces, I would feel comfortable starting with the "belief"
that there is a quantity called energy, and a Principle of Least
Action to go with it. I *want* to be able to use it to create a
notion of force as the gradient of an energy, after which I would be
happy to then proceed with the traditional F=m*dp/dt to solve
problems when it is easier to do so. As someone mentioned in the
thread back in Aug, arming oneself with the energy concept, and
adding conservation laws, which have a basis in fundamental
symmetries, allows one to go a long way. But my notions outlined
above make me think that things are quite "messy." And perhaps they
are unavoidably so.
So, wherefore the Principle of Least Action? Are my observations
above off-base? Is there a treatment out there that starts with the
Principle, delineates branch points for different physics
disciplines, and systematically but comprehensively addresses the
so-called shortcomings?
Stefan Jeglinski
[1] I have purposely left out quantum mechanics, even though one sees
more discussion of the Principle there, as the Hamiltonian is more
"developed" than in classical mechanics. But perhaps this is the key?
Perhaps the only way to clear this up is to make QM the starting
point for Least Action, and derive paths to the other disciplines
from it?