[Phys-l] symmetries of the Lagrangian

[John Denker <jsd@av8n.com>]

On 01/20/2010 09:44 PM, Stefan Jeglinski wrote:

>  I like the idea pedagogically of an argument that doesn't 
> resort to blatant/circular assertions about the form of KE. However, 
> Landau/Lifshitz do not expound further either, beyond these few 
> words. I think I understand the second part, essentially that the 
> isotropy of space cannot support a preferred direction, but it's 
> unclear to me how the homogeneity of space and time lead to L being a 
> function of v alone. 

It's easy to demonstrate that the L&L argument is
bogus. 

As the saying goes, it's bad luck to prove things
that aren't true.

Their argument sorta proves one thing, but they
claim to have proved something else.  I would be
quite willing to believe some sufficiently-careful 
statements about what terms *cannot* be in the 
Lagrangian ... but to leap from there to statements 
about what terms *must* be in the Lagrangian is 
just crazy.

Specifically, I am willing to believe that there
cannot be a term containing just plain naked v all 
by itself.  That would have the wrong symmetry.
On the other hand, the electromagnetic Lagrangian
contains a term in j•A which is first order in j,
which is tantamount to being first order in v.  A
term in plain naked j would be disallowed for the
same reasons plain naked v would be disallowed.

There are pretty good symmetry arguments why the
Lagrangian density should be a scalar ... not just
a 3D Galilean scalar, but also a 4D Lorentz scalar.

We must not leap from there to any notion that the
Lagrangian will only contain terms that are second
order in v.  As others have pointed out, terms in
|v| are allowed by symmetry, as well as just about
any hypothetical function of |v|.  Also terms in
v•A are allowed;  this is not a vague hypotheses, 
but known good physics.

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

And now for a book review:  If you are ever reading
Landau and Lifshitz and you come to a passage that
you don't understand, you reeeally need to consider
the possibility that the passage is wrong, misleading,
or at best highly open to misinterpretation.

In particular, their "proofs" are notorious.  If you're
lucky, they state a conclusion that is true enough as 
a matter of fact, but then "prove" it using a "proof" 
that proves nothing of the sort.

If you're not so lucky, the conclusion is not entirely
true, but instead is subject to all sorts of provisos
that they don't bother to mention.

For these reasons I consider the whole series of books
to be a pedagogical disaster area.

====

On happier note, if you want to learn about classical
mechanics -- and learn a lot of other good stuff along
the way -- I recommend
  Gerald Jay Sussman and Jack Wisdom with Meinhard E. Mayer  
  _Structure and Interpretation of Classical Mechanics_
  http://mitpress.mit.edu/SICM/

The whole thing is available for free on the web, but
once you get into it you may well decide to buy the
hardcopy.  The table of contents is at:
  http://mitpress.mit.edu/SICM/book-Z-H-4.html

Modern.  State of the art.

Lucid.

Abounding in good worked examples.

You'll get far more out of this book than you possibly
could get out of L&L.