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[Phys-L] Re: Energy is primary and fundamental?



Stefan Jeglinski wrote:
...

Having said that, the Principle seems to have shortcomings.

I'm not convinced it has any shortcomings.

2. A criticism leveled at the classical mechanics version of the
Principle is that although it can used to derive Newton's Laws (or at
least the 2nd one), one needs Newton's Laws to do so.

This is not an accurate criticism. You can rigorously derive all
of classical mechanics from PoLA. Been there, done that. In particular,
on several occasions I was faced with situations where I didn't know
for sure what the classical mechanics was ... so I sorted it out by
relying on the Lagrangian.
For example, suppose you think an electrical LC oscillator is analogous
to a mass on a spring. Surely it ought to be, but what are the details?
Series or parallel circuit? Is L the mass, or is C the mass? Are
you sure? .... I was able to answer all those questions by choosing a
coordinate, writing down the Lagrangian, and turning the crank.

"The Lagrangian knows all and tells all." In particular, given
the Lagrangian, it will *tell* you what quantity is dynamically
conjugate to the variable you have chosen ... and then you can
find the Hamiltonian in terms of the Lagrangian. The converse
is not true; knowing the Hamiltonian is not sufficient to find
the Lagrangian.

Also note that the Lagrangian density is a relativistic invariant.

So, wherefore the Principle of Least Action? Are my observations
above off-base? Is there a treatment out there that starts with the
Principle, delineates branch points for different physics
disciplines, and systematically but comprehensively addresses the
so-called shortcomings?

...
I have purposely left out quantum mechanics, even though one sees
more discussion of the Principle there, as the Hamiltonian is more
"developed" than in classical mechanics. But perhaps this is the key?
Perhaps the only way to clear this up is to make QM the starting
point for Least Action, and derive paths to the other disciplines
from it?

If you want action, you need the Lagrangian, not the Hamiltonian.

And yes, you can do QM in terms of the Lagrangian. The phase of
the wave function goes like exp(i S / hbar), where S is the action.
There's a reason why h is called "the quantum of action".

Indeed, although the Hamiltonian is king in nonrelativistic QM,
as soon as you start doing relativistic QM you're in Lagrangian
territory. FWIW I get 100,000 hits from
http://www.google.com/search?q=superstring+lagrangian

You need to think about action in general, not just least action.
The *least* action emerges from the general expression in the
classical limit, typically via a stationary-phase argument to
find the "most classical" contribution.

To understand this,
-- Start with Feynman volume II chapter 19 ("The principle of
least action") which is quite readable and in the usual
Feynmanesque way covers a lot more than just the PoLA.
-- Then read Feynman _QED : The Strange Theory of Light and Matter_
which is easy to read i.e. aimed at a minimally-technical audience.
-- Then read Sussman & Wisdom
_Structure and Interpretation of Classical Mechanics_
http://mitpress.mit.edu/SICM/book-Z-H-3.html
I quote: "This book is dedicated, in respect and admiration, to
The Principle of Least Action"
Note that the entire book is readable online for free. But I liked
it enough to buy the hardcopy anyway.
-- Finally read Feynman and Hibbs _Quantum Mechanics and Path Integrals_
which derives more-or-less everything (including thermodynamics!)
starting from Lagrangians and actions. This is, alas, not easy
reading. Some of my friends who have Nobel prizes have been known
to wince at the sight of this book. Also beware that it has typos
galore, which contributes to making it hard to read. But the ideas
are there, and clearly expressed. It took me about 180 hours to
work through this book, doing the exercises et cetera. (That
was an hour a day for six months.) I'm not sure I would want to
calculate anything in detail using path integrals, but the
qualitative insights that I have gotten by thinking about things
that way have been priceless. In particular, it gave me a "feel"
for what QM _exchange_ is, as a physical (not just mathematical)
process. That in turn gave me a "feel" for what Bose-Einstein
condensation is, and how that relates to superfluids and
superconductors.