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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:
......
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.