Re: [Phys-l] local conservation (was: non-grady ...)

[John Denker <jsd@av8n.com>]

On 02/20/2008 01:56 PM, Steve Highland wrote:
> > To me, conservation of energy is on exactly the same footing as
> > conservation of charge.  Energy is somewhat abstract, but so is
> > charge.  Energy can be viewed as "stuff" that can flow across
> > the boundary, and so can charge.  Energy can be formalized as a
> > completely abstract bookkeeping exercise, and so can charge.
> [John Denker]
>
> I'd like to see how electric charge could be reduced to just bookkeeping.  I
> confess to being bothered (in a way I can't quite put my finger on yet)  by
> your claim above.
>
> I still want to cling to the idea that electric charge is real, tangible
> 'stuff' that I can hold in my hand.  Energy seems different, especially when
> you can freely label any state as 'zero energy' and get the same description
> a a system's behavior.  How can energy be real 'stuff' if you can
> arbitrarily say how much you have to start with?

By way of analogy: Altitude is as "real" as anything could be, 
yet it has a gauge invariance much like energy does.  The choice 
of gauge has no observable consequences.  This doesn't make it
any less "real".

Actually, this whole line of discussion as to whether charge is 
more "real" than energy has no observable consequences.  I don't 
think it is helpful (let alone necessary) to pursue such questions.

If you think charge is not somewhat abstract, please explain the
relationship between one mole of electrons and one mole of muons.
It seems to me that practically every "tangible" detail is different, 
except that the /charge/ is the same.

Formally, there is an all-purpose definition of local conservation.
It can be visualized in terms of continuity of world lines.

> I recall that conservation of charge is a direct consequence of Maxwell's
> equations, but read yesterday that energy conservation (in the first law of
> thermodynamics sense) is a deep statement that contains *more* than what can
> be derived from Newton's laws.

Let's be careful.  

It is a two-line calculation to show that the Maxwell equations 
/uphold/ conservation of charge.  That is not, however, a proof
that charge is *always* conserved, because we are not rash enough
to think that the Maxwell equations govern all of physics.  For
example, the radioactive decay of a neutron is observed to conserve
charge, but I wouldn't consider that a "direct consequence of
Maxwell's equations".

The Maxwell equations are known to be incomplete.  In particular,
you can't do much of anything with them unless you tack on the
Lorentz force law ... and it is obvious that the force law doesn't
include gravitation or nuclear interactions.


> "The basic problem is that the work-energy theorem is really a dynamical
> relation, derived from Newton's second law, and cannot be a truly general
> energy statement. There are three independent conservation relations in
> classical physics. The first is the equation of continuity or conservation
> of mass. The second conservation law, that of conservation of momentum, is
> the grand generalization underlying Newtonian dynamics. If the general law
> of conservation of energy in classical physics could be derived from the
> dynamical equations, it would be a tautology and would not constitute a
> third independent condition, as it actually is. The law of conservation of
> energy cannot be derived. Like the other conservation laws, it is arrived at
> by induction from limited observations and is ultimately accepted because no
> violations are observed."
> [Arnold Arons
> Development of energy concepts in introductory physics courses
> American Journal of Physics, Vol. 67, No. 12, pp. 1063­1067, December 1999]

As usual, about half of what Arons says is wrong.  There are not
"three independent conservation relations" if you count charge.
The various conservation laws were not discovered all at one time,
so it is reasonable -- and trivial -- to surmise that one of them 
must have been the first to be discovered.  If you consider Newton's 
third law as a crude expression of conservation of momentum, then 
this is the longest-known conservation law ... but who cares?  This 
is just a historical accident of the most trivial sort.

It is plausible to conjecture that if Galileo had lived a little
longer, conservation of energy would have been asserted before
conservation of momentum.  He clearly understood special cases of 
conservation, 
  http://www.av8n.com/physics/loop-de-loop.htm
and could plausibly have lept to the full generalization eventually.

For that matter, giving Newton credit for understanding conservation
of momentum /as a conservation law/ is a bit of a stretch.  Newton,
Leibniz, and their contemporaries wrangled endlessly over "vis viva"
and couldn't agree on whether "the" conserved quantity was what we
nowadays would call momentum or would call energy.  And again, who
cares?  Nowadays we know what the conservation laws are, and fussing 
about whether they are logically "independent" of other things we 
know is of little value.  They can be made independent or not, depending
on exceedingly fine details of how other laws are formulated, which
comes down to questions of taste in the end.  In any case, all this 
is well outside the scope of an introductory course.

> Doesn't this put energy on a very different footing from electric charge?

No.

There is an all-purpose notion of local conservation, and it covers 
charge, energy, momentum, lepton number, etc. just fine.

I do not want to argue about taste, or about style, but there is a
school of thought that /starts/ from the great conservation laws
and develops the rest of physics from there.  Force can be explained
in terms of energy at least as conveniently as vice versa.

On 02/19/2008 08:22 PM, LaMontagne, Bob wrote:

> > Simply starting with energy consevation out of the blue and ignoring
> > its link to Newton's 2nd through KE seems too authoritarian - even
> > though  it's a valid approach to physics.

I've never advocated an "authoritarian" approach.

All too commonly, F=ma is deplorably "authoritarian", introduced on the
authority of Sir Isaac Newton.  Yes Sir!  Even if energy were somewhat 
authoritarian, it wouldn't be worse.

But to turn the conversation onto a more positive track, why should
either of these notions come "out of the blue"?

The logical foundation for asserting conservation of energy is _at least_
as firm as the foundation for asserting F=ma.  In both cases there are
innumerable observations that can be used to make the assertion plausible.
This is a scientific (non-authoritarian) approach: We learn some things
by observation, then we write down a law that summarizes what we have
learned.