Re: Can energy conservation be derived from Newton's motion laws

From: Daniel E. Platt (DanP57_at_optonline.net)
Date: 01/06/05 

Date: Thu, 6 Jan 2005 22:11:30 +0000 (UTC)

Strong_Field wrote:
> "Dan Platt" <DanP57@ispwest.com> wrote in message
> news:cqpqkq0hvj@enews3.newsguy.com...
>
>
>>Grad is usually definable this way (x, dx vectors; s, V scalars):
>>
>>V(x + s dx) = V(x) + s grad V . dx + O(s^2).
>
>
> This seems to corroborate what I wrote initially. Ignoring your functio=
ns s
> and O, I don't know what they are, your expression simplifies essential=
ly to
> dy/dx = grad V. Does this definition say more than the definition giv=
en in
> the other post as a vector pointing in the greatest increase and having=
 the
> magnitude of change in velocity?

This was a Taylor (power) series in s, easier than a power series in
mixed dx components. The O() refers to powers of s^2 and higher (ie,
some number times s^2 + another number times s^3 +...). It did look
like the kind of thing you said you wanted, along with a method to get
there. However, it is not always possible to extract such a power
series (the degenerate perturbation series for matrices is an example).

>
>
>>... the idea of conservation of energy is a fundamental (distinct from
>>basic) physical idea. Perhaps it is more meaningful to ask what kind o=
f
>>force satisfies conservation of energy given Newton's laws. If
>>conservation of energy is a dominant physical mode of behavior that
>>penetrates into thermodynamics, etc, then all of the important forces
>>that will show up in the application of Newton's laws will have a
>>gradient-like behavior.
>
>
> I think you are trying to say that conservation of energy is the fundam=
ental
> axiom of physics. If so I agree.

I would not even quite say it is an axiom. Physics is descriptive, with
lots of loose pieces that don't quite fit together very well in all
places (though there has been lots of progress in finding ways to make
them fit).

> The equation itself is a statement of t=
he
> conservation of energy. Everything else is derived from the assumption =
that
> perpetual motion is impossible.

You *can* derive an expression of the 2nd law of thermodynamics that way
(well, you can do it more precisely in terms of heat engines). There is
no way to construct a probe into the microscopic world without injecting
some kind of statistical assumption. It doesn't help that the
statistical pictureS have multiple forms. For example, the picture of
an ensemble involves multiple instances of a single experiment. But we
see thermodynamic equilibration over one single experiment for a single
system. Then there's Birkhoff's thm, that states that systems with
certain characteristics will always show a time average close to the
ensemble average for all functions... but usually, we don't look at all
functions, rather we look at a very special set of macroscopic
functions, and the time we sample is much shorter than the time required
for a system to sample most of phase space (which is often rather larger
than the age of the universe). So, for some very special but very
interesting macroscopic functions, we see thermodynamic behavior; for
microscopic variables, the quantum scattering processes look just like
individual events. As a point of interest, you don't have to get too
small to run into a circumstance where this type of thing starts to
break down... Mitochondria do a very important biological task in our
cells, converting ADP to ATP, mediating the reaction with the transfer
of protons (H+ ions) through transmembrane enzymes called ATPase. The
question is, what is the charge distribution like near the membrane
where this happens? It turns out that the density of the charge
carriers is smaller than the Debye screening length you get if you
assume that chemical potential and smooth particle densities are valid.
  The mistake is that you only get maybe 6 loose protons in a
mitochondrion at a time, and the assumption of thermodynamic diffusion
at that scale doesn't quite work.

> Let's think about it topologically.=
 In
> topology equivalent objects can be transformed into each other. It is o=
nly
> convention that the shape A is the master shape. Someone will say no th=
e
> shape B is the master shape. Similarly in physics there is no master
> equation that can be identified as such. In physics any equation can be
> derived from the other. Whatever shape you give to it your topology sta=
ys
> invariant. That=92s the energy conservation. The topological invariant.=
 Which
> is the equation itself. Because whatever its =93shape=94 is, the equati=
on is the
> invariant. Therefore in physics the equation itself is the definition o=
f
> energy conservation. As such you cannot derive it from any other equati=
on.
> You can only use energy conservation to prove whatever you want. This a=
lso
> shows that deriving energy conservation from Newton amounts to making
> algebraic transformations.

Energy conservation is intimately connected to time translation
invariance (ie -- the description of the mechanics of a system is
independent of when the experiment is performed). It also emerged in a
distinct statement in the realization that energy did not disappear in
non-conservative mechanical systems -- it was converted to/transfered as
"heat." These distinct ideas were re-connected through statistical
mechanical considerations. The notion of energy conservation suggests
energy transfer; it isn't just mechanical things that can carry energy,
so can fields (electromagnetism). This connected to issues of
force-at-a-distance (instantaneous energy transfer) that would show up
as a problem, later. One face of protecting the form of the statement
of electrodynamics as basic physical law was the realization that time
and space both transformed covariantly for different observers
(otherwise, you have an absolute space again -- which Newton's laws
provided no means of detecting by themselves). If so, then energy and
momentum also transform in a similar way as space and time. Also,
formulations of fluid energy density transfer are identical to fluid
mass transfer... you canot tell the difference.

>
> And, conservation of energy is not a =93physical mode of behavior.=94 I=
t is an
> axiom. There is nothing physical about it.
>

Rather, it is something that gets discovered and re-discovered over and
over again in new contexts, usually with a face that is not easily
reconciled with prior pictures. The part that is "physical" is that it
must be connected to empirical experience, or else it is simply a
mathematical diversion.

>
>
>>Humans are trying to describe the behavior of things that would be doin=
g
>>what they do regardless of whether humans were there to describe them.
>>To some extent, the notion of physical law is a human construct, which
>>is subject to change as our understanding improves.
>
>
> You are talking about a philosophical and epistemological concept of
> =93physical law.=94 I agree that=92s a human construct. But that
> discussion is not relevant to =93Newton=92s laws.=94 Because
> Newton=92s Laws are not =93physical laws.=94 They may be laws of
> Newtonian physics, but they are not laws of nature in the sense you are
> thinking. Newton was a serial definer. He had the authority to establis=
h
> his countless definitions as laws of nature in physics. If you look at
> the Principia, they are called axioms. In other words, they are
> definitions Newton chose to make. By tradition we call them
> =93laws=94 out of respect for Newton. As long as we know that they
> are laws with an asterisk, I guess there is no harm done to science.
>

It doesn't take much authority to make an axiom. You have to start
somewhere. The notion of physical law is a harder problem: your
description has to conform to observation to the best of your
experimental capability. The interplay between math and description
occasionally reveals striking new insights into the relationships
between ideas (symmetry and conservation, for instance) that otherwise
would not be obvious. I think the exciting spots are where things don't
fit together very well... that's likely where some understanding is
going to emerge when those issues are resolved.

Dan

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