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# Re: [Phys-l] Landau on Lagrangian

• From: John Denker <jsd@av8n.com>
• Date: Mon, 25 Jan 2010 10:20:44 -0700

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

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.

Agreed. Well said. Thanks.

And just to make sure everybody is on the same page:
a) It is one thing to decide in advance we want a
particular equation of motion, and
b) It is quite another thing to decide in advance
on a set of symmetries and to pretend that suffices
for deriving the equation of motion.

On 01/25/2010 09:09 AM, Hugh Haskell replied:

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.

OK.

But what is the right
Lagrangian? Evidently, it is whatever will give us EOMs that conform
with what we observe.

OK.

How we get there is pretty much irrelevant.

It is irrelevant up to but *not including* the point
where we (or L&L) misrepresent how we got there.

Any class has multiple goals. The primary, fundamental,
and overarching goal should be for students to learn to
think clearly. Learning this-or-that detail about the
equations of motion is secondary.

Telling students that symmetry principles suffice to
derive the equations of motion -- when that is not in
fact true -- is a huge step in the wrong direction.

If you want to say "we pulled this Lagrangian out of
a hat and we like it because we know by means of 20/20
hindsight that it will generate the nonrelativistic
equations of motion for a point particle" ... that's
fine. If you want to say that "it has some symmetries
that any physical Lagrangian ought to have" that's OK
too.

We
can guess,

OK.

or we can impose some abstract symmetries,

Necessary but *not* sufficient.

I haven't read the whole L&L passage, and I'm not
particularly interested in doing so ... but in any
case it is clear that if people interpret the passage
as saying that symmetry principles suffice to derive
the Lagrangian, then either the passage is wrong or
the interpretation is wrong.

The wrongness should be obvious from the fact that
the Galilean symmetries that we are discussing are
a subgroup of the Lorentz symmetries of special
relativity. We know that relativistic mechanics
is different from non-relativistic mechanics, so
the idea that Galilean symmetries lead to one and
only one equation of motion is dead on arrival.

.... Neither Lagrange nor Newton are fundamental to nature.

Agreed!

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.

No, it does not. Symmetries are not sufficient to
deduce the equations of motion.

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, ....

Agreed. When expressed as a guessing game, validated
by 20/20 hindsight, that's entirely correct. That is
how it is often presented.

=================================

Symmetries *are* related to the equations of motion.

Once you know the Lagrangian, then Noether's theorem
says that for every continuous symmetry of the action,
there is a corresponding conserved quantity. But
once again, you have to know the Lagrangian first.

This is what we should be discussing! Noether's
theorem is real physics, beautiful physics.

Exegesis of some Landau/Lifshitz snafu, not so much.