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Re: [Phys-L] "conservative force" --> misnomer --> misconception



On 01/15/2014 10:44 AM, Carl Mungan wrote:
It is very common to find people talking about conservative
and non-conservative forces in the same breath as conservation
of energy.

Don't they mean "conservation of mechanical energy"?

Good question. I don't know what they mean.

Since the connection between conservation of energy and
so-called "conservative forces" is so widespread, we should
consider the possibility that there is some genuine reasonable
meaning, and I just haven't figured out what it is.

Obviously it depends on how you define "mechanical energy".
Let's give it a try:

a) Consider the potential energy of a brick in the earth's
gravitational field. Surely this must be considered part
of the "mechanical energy". Somebody (IIRC Clausius?)
touted this as the operational /definition/ of energy.
b) Consider the potential energy of the moon in the earth's
gravitational field.
c) Consider the potential energy of the earth in the moon's
gravitational field.

The problem is, (b) and (c) are the same thing, and if you
include them both, you're double-counting, and you get the
wrong answer. Furthermore, if (b) and/or (c) is wrong,
then (a) must be wrong also. So in fact they are all three
wrong. The only way out is
d) The gravitational energy is in the field, not "in"
either of the gravitating objects.

We can recognize (a) as an approximation to (d). This is
an exceedingly useful approximation, highly accurate for
ordinary bricks under ordinary classroom conditions.

HOWEVER ... At the fundamental conceptual(*) level, this
prevents me from accepting the idea that mechanical energy
is conserved if-and-only-if the force is the gradient of
some potential, i.e. the force is grady, i.e. (loosely
speaking) it is a so-called "conservative force".

Specifically: Let's switch from gravitational energy to
electromagnetic energy. As discussed above, we must consider
the field energy to be part of the "mechanical energy". OTOH
if we do that, the mechanical energy is conserved even if
the electromagnetic force is non-grady, e.g. in a betatron.

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

To summarize: I have tried to find a connection between
conservation of energy (or some subset of the energy) and
grady force-fields, i.e. the so-called "conservative forces".
There are two possibilities:
a) There is a connection, but I haven't thought it through
properly.
b) The folks who claim there is a connection haven't
thought it through properly.

At the very least, I can object on pedagogical grounds that
the usual arguments in favor of the alleged connection are
exceedingly sloppy. I haven't seen much beyond cursory hand-
waving, proof by bold assertion, and proof by pun.

If anybody has a better explanation of the alleged connection,
please let us know!