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*From*: David Bowman <David_Bowman@georgetowncollege.edu>*Date*: Mon, 25 Jan 2010 17:58:47 -0500

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.

Yep.

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

vice-versa.

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.

Hugh

David Bowman

**References**:**[Phys-l] Landau on Lagrangian***From:*Stefan Jeglinski <jeglin@4pi.com>

**Re: [Phys-l] Landau on Lagrangian***From:*"Bob Sciamanda" <treborsci@verizon.net>

**Re: [Phys-l] Landau on Lagrangian***From:*Stefan Jeglinski <jeglin@4pi.com>

**Re: [Phys-l] Landau on Lagrangian***From:*David Bowman <David_Bowman@georgetowncollege.edu>

**Re: [Phys-l] Landau on Lagrangian***From:*"Bob Sciamanda" <treborsci@verizon.net>

**Re: [Phys-l] Landau on Lagrangian***From:*David Bowman <David_Bowman@georgetowncollege.edu>

**Re: [Phys-l] Landau on Lagrangian***From:*"Bob Sciamanda" <treborsci@verizon.net>

**Re: [Phys-l] Landau on Lagrangian***From:*David Bowman <David_Bowman@georgetowncollege.edu>

**Re: [Phys-l] Landau on Lagrangian***From:*"Bob Sciamanda" <treborsci@verizon.net>

**Re: [Phys-l] Landau on Lagrangian***From:*David Bowman <David_Bowman@georgetowncollege.edu>

**Re: [Phys-l] Landau on Lagrangian***From:*"Bob Sciamanda" <treborsci@verizon.net>

**Re: [Phys-l] Landau on Lagrangian***From:*Hugh Haskell <haskellh@verizon.net>

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