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Re: ENERGY BEFORE Q

At 10:11 PM 10/18/01 -0400, Ludwik Kowalski wrote:

Model 1 --> idealized textbook world without friction.
Model 2 --> a little more realistic world with dissipative forces.
Model 3 --> a more realistic world with engines, etc.

Possible minor refinement:

Model 2a --> a little more realistic world with nonconservative forces, but
no dissipation (no entropy production). Example: Charged particle in
time-varying magnetic field.

Model 2b --> a little more realistic world with out-and-out dissipation,
internal degrees of freedom, et cetera. Example: friction.

> I stand by my assertion: Defining work in terms of
> potential energy [this was NOT my suggestion] would
> be a blunder. Defining potential energy [as above] in
> terms of work would be an even bigger blunder.

I am puzzled. I do not want to go from Model 1 to Model 2
without resolving the above issue. The three lines above
represent the very essence of Model 1. Why were they
labeled as a "big blunder."

We need to be clear about what sort of "work" we are talking about.

1) When I spoke of "big blunder" I was thinking of the conventional
thermodynamic notion of work, such as shows up in the unfortunate equation
E = W + Q
I have vehement objections to this equation and to the ideas it expresses,
as explained in some detail at
http://www.monmouth.com/~jsd/physics/thermo-laws.htm
These objections are relevant to this thread because "Q" appears in the
Subject: line, and because model 2 explicitly mentions dissipation.

2) If OTOH we are talking about work in terms of F dot dx, and if we are
willing to impose a number of restrictions and assumptions, and if we don't
mind some slightly circular arguments, then we can make some
pedagogically-useful connections to potential energy.

3) There is a big difference between
-- _defining_ energy in terms of work, as opposed to
-- verifying an equality between work and certain pieces of the energy
under certain circumstances, based on other definitions.

4) There is an intimate connection between force and momentum. IF (big if)
you know the relationship between momentum and kinetic energy, you can
connect work to kinetic energy. (Note we are calculating kinetic energy,
not potential energy. Also note we are calculating the KE, not defining
it.) Then if you want to connect this with potential energy, you have to
make several major assumptions:
a) You need to _assume_ that KE+PE=constant.
b) You need to _assume_ that the force field is conservative, so that it
is possible for there to be a PE function that depends only on position.
c) We are still assuming we know the KE as a function of momentum.

Let us discuss each of these assumptions:

A) I've seen way too many textbooks that claim they have "proved"
conservation of energy, when in fact they just assumed it, as assumption
(1a). The alleged proof is blatantly circular.

If you are going to _assume_ KE+PE=constant, why not be up front about it?

B) Introductory texts typically restrict the discussion to conservative
force-fields. But this (like the previous item) seems circular to
me. Circles within circles. Assuming the force-field is conservative is
more-or-less tantamount to assuming it is the gradient of some
potential. So defining the potential in terms of a conservative force
doesn't tell us anything beyond what we just assumed.

And what if the force-field isn't conservative? I guarantee you that your
office is filled with non-potential voltages, due to time-varying magnetic
fields. Can you tell me what is the potential energy of an electron under
such conditions? The question as stated is unanswerable, even though F dot
dx remains perfectly well defined. If we ask different questions, it
becomes possible to deal with non-potential voltages, but you need
additional variables (such as the current) and a lot of additional
machinery -- and yet more assumptions.

So if you want to proceed to model 2, don't underestimate the amount of
effort involved. Don't claim you've dealt with it when you haven't.

Note that we are talking about force-fields (such as a time-varying
magnetic field) that are 100% physical and which fully uphold the law of
local conservation of energy. They just do it in a complicated way.

I can't imagine any way to deal with Model 2 correctly in an introductory
course. I'm sorry to be so negative, but we can't simplify the physics
just by wishing.

In a more positive vein: I would be tempted to say something like this:
We are going to assume that energy is conserved. This
assumption is based on a mountain of experimental evidence
(that we will discuss bit by bit as we go along) and
theoretical considerations (that we're not going to
discuss in this course). If we divide the energy into
KE and PE, then KE+PE=constant. We will from time to
time verify that other things we know, such as the force
laws, are _consistent_ with KE+PE=constant. However, we
are not going to use the force laws to "prove" that
KE+PE=constant. Doing so would be a fool's errand,
especially if there are time-varying magnetic fields
around. If you want to see how to derive the force
laws from energy considerations (not the other way
around), read Feynman's lecture on the Principle of
Least Action.

C) You can't blithely assume you know the relationship between momentum and
kinetic energy. For example, a macroscopic system can hide KE in internal
degrees of freedom, such as thermal kinetic energy and/or spinning
flywheels. In contrast, you can't hide momentum; if you have momentum
your center-of-mass is going to be moving, and there's nothing you can do
about it.

5) This isn't 100% relevant to the question Ludwik asked, but it may cast
an interesting sidelight on these issues: It turns out that the laws of
nature can't tell the difference between KE and PE! The distinction arises
only as a consequence of choices that humans make when analyzing the
system. In particular, consider an LC circuit. There is energy in the
inductor, and energy in the capacitor. Which is KE, and which is PE? It
doesn't matter! You can choose it either way, depending on whether you
choose flux as your "coordinate" or charge as your "coordinate".

4) It would certainly be wrong to define a "frictional potential"
as work done again friction (or air resistance, etc.) because
that work depends on the length of the path and not only on
locations if the initial and final points.

Yes.

But why should potentials
associated with conservative forces not be defined in terms of
these forces?

If you change the verb from "defined" to something like "illustrated" or
"explained" or "introduced" it would make me happier. The "definition"
seems circular as explained above.

How should the above three potentials introduced
in the first physics course?

Good question. There are numerous true things we could say (and even more
numerous untrue things we should avoid saying).

The question asks about "introducing". Introducing examples is
easy. Proving the general case is hard. Introductory examples include
connecting certain force laws to the corresponding potential energy:
F(spring,particle) --> PE(spring,particle)
F(gravity,particle) --> PE(gravity,particle)
and you can equally well start from an energy principle and go the other
direction
PE(spring,particle) --> F(spring,particle)
PE(gravity,particle) --> PE(spring,particle)
and the choice of direction is mostly a matter of taste.

The tricky bit comes when we try to generalize, to prove the general rule
F(particle) --> PE(particle) in general

If you prove that, you've got a slight problem: When we get to Model 2,
with non-potential voltages, the last-mentioned "general rule" simply is
not true. (It's bad luck to prove things that aren't true.)

As far as I can see, a first physics course cannot possibly erect the
machinery to give a microscopic description of conservation of energy in a
force field that is not derived from a potential. The simplest example in
this category is a time-varying magnetic field, which is
non-dissipative. Examples involving dissipation, entropy, and friction are
vastly more complicated.

To say it again: Students are surrounded day in and day out by forces that
aren't derived from any potential. Elementary physics gives them the tools
to ask how this relates to conservation of energy, but does not provide the
tools for answering the question.

This situation is not unprecedented. By the time students leave high
school, they have seen a number of problems where the question can be
stated in elementary terms, but the answer cannot be explained in
elementary terms. Examples include
-- The four-color map theorem.
-- Non-trisection of an angle with ruler & compass.
-- Fermat's last theorem
(which I suppose should be called Wiles's theorem).

This is why physics teachers resort to explaining conservation of energy by
analogy to other conserved quantities: Dennis has a conserved number of
blocks; money is conserved in ordinary retail transactions. These
analogies don't prove anything, but they get the point across. And I think
that's the best we can do in an elementary course.

If anybody can do better I'd be delighted to hear about it.....