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Re: [Phys-l] Landau on Lagrangian

Regarding Hugh's insights:

When we know the answer before we start "deriving" it, it's hard
to tell just how much our prior knowledge has influenced the
result we get. And I think that's what's going on here. As I read
the discussion on this thread, I see the Lagrangian approach as a
very general one that asserts that, if we find the right
Lagrangian, it will give us the right equations of motion. But
what is the right Lagrangian? Evidently, it is whatever will give
us EOMs that conform with what we observe.

I agree 100%.

How we get there is pretty much irrelevant. We can guess, or we
can impose some abstract symmetries, or we can take our cue from
what we already know (Newton's laws) and construct an appropriate
Lagrangian (which is the way Lagrangian dynamics is usually
introduced in advanced mechanics courses). We are not, as I see
it, assuming the form of kinetic energy in the Lagrangian--we are
assuming the appropriate symmetry and the form of kinetic energy
follows, just as it does from Newton's approach.


David Bowman's approach described in this thread is to impose some
observed symmetries of nature on the Lagrangian and see where it
leads us.

Actually, I was just trying to relay what I had remembered was the gist of L&L's argument. I wasn't trying to make any claims about how much that argument has built into it the form of N2 in a hidden implicit way. It is entirely possible that, taken together, 1) Hamilton's Principle, 2) the collection of the imposed symmetries (i.e. invariance under the inhomogeneous Galilei group), and 3) the 2nd order requirement of the EOM are fully mathematically equivalent to N1 & N2. My purpose was only defending the argument L & L against the charge that it was an *explicit* assumption of N1 & N2. It very well may boil down to an *implicit* such assumption, which I would not care to dispute at any length.

If we choose the right symmetries (that is, all of the actual ones
that we observe that seem to apply to nature), then we find that
the Lagrangian must depend on v^2, and this then leads us to
Newtonian EOMs. We could have started with Newton's laws (which
already have those symmetries included in them) and derived the
appropriate Lagrangian, which would be the same one we found
directly from the symmetries.

Should this be a surprise? Is it circular? Well, no and yes. It
is circular, but not in the slightly pejorative sense that several
here have implied. Neither Lagrange nor Newton are fundamental to
nature. What is fundamental is the symmetries. If we go one way
around the circle from the symmetries, we get Lagrange, and that
leads to Newton. If we go the other way we get Newton and that
leads to Lagrange.

What that tells me is that the Lagrangian method is good if we are
clever enough to guess the Lagrangian (i.e., the one that leads to
equations that predict what is actually observed). It seems rather
apparent to me that when we find the correct Lagrangian, we get
Einstein, or Heisenberg/Schroedinger, or Dirac/Feynman, and

Am I heading for (or already arrived at) La-La Land, or am I
assuming too much here?

I don't know what others may say, but, I for one, like what you wrote here.


David Bowman