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# Re: [Phys-l] Landau on Lagrangian

At 08:32 -0500 01/25/2010, Bob Sciamanda wrote:

OK
The mathematical connection between the least action principle and the
Lagrange equations which generate the "EOM" is a purely mathematical logical
development, with no input from Newtonian mechanics to further specify the
Lagrangian L ( indeed it is often applied to non mechanical, purely
mathematical problems, eg geodesics).
You have added some mathematical requirements to be imposed upon L when this
development is to be applied to mechanical motion; ie, to generate Newtonian
dynamics.

These requirements can be accepted as also "purely" mathematical, with no
overt input from the conclusions of Newtonian mechanics (this may be
arguable).
But even If I grant you these requirements and accept that they imply that
the mechanical L = L(v^2), you still have to show that this implies:
KE(translation) is proportional to v^2.

This is a statement of Newtonian dynamics and cannot be deduced from "pure"
mathematics. If it could be so deduced, Newtonian dynamics would be (at
least in part) a metaphysical necessity and not just a falsifiable
conclusion of empirical physics. Indeed the very concept of KE is defined
within Newtonian dynamics and is not even a part of the language of the
purely mathematical calculus of variations.

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. 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. 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 vice-versa.

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

Hugh

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