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

From: Strong_Field (strong_field_at_hotmail.com)
Date: 01/04/05 

Date: Tue, 4 Jan 2005 20:06:01 +0000 (UTC)

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1104169210.571937.84550@f14g2000cwb.googlegroups.com...
> Strong_Field wrote:
>
> It is important not to confuse
> notation with meaning. The whole derivation can be written out in plain
> English without notation from calculus. However, every derivation
> starts with some given assumptions. Discussion of meaning is usually
> reserved for these assumptions.
>
“Newton’s laws” and “Energy conservation” appears to be two different names
for the same thing. In any derivation, there is the first statement X, and
there is the last stament Y. If your derivation starts from X and ends with
the expression Y, then the expression Y must be hidden in the expression X.
If Y is not hidden in X and you could find Y starting from X you would be
violating conservation laws. You would be creating something from nothing.
Equations themselves obey the rules of thermodynamics. An equation is
nothing more than a definiton of conservation rules. Otherwise you cannot
express the laws of thermodynamics with physics equations. So, in my
opinion, what physicists call a “derivation” is nothing more than an
algebraic simplification process. To claim otherwise would be against the
laws of physics. Corrections, comments welcome.

> The physics content is contained in the assumptions. They can be
> summarized as (1) Newton's second law and (2) only conservative forces.
> The mathematica definition of a conservative force has already been
> stated. Are you looking for a justification of this definition?

To a disinterested observer who does not know the conventions and traditions
of physics but is familiar with mathematical notation, what you call “Newton
’s second law” is simply a definition. “Newton’s second law” makes the two
symbols “F” and “a” synonyms. There is no reason to write the superluous
symbol “m” which will cancel at the next step. (Or if you wish we can set it
to unity.)

F == a is in itself nothing more than saying that in your equations you may
write “F” or if you think it looks better you may write “a” instead. There
is no physics content in F == a.

It is an old physics tradition to call “acceleration” “force” and I have
nothing to argue with that. Physicists also like to call old definitions
coming down from the 18th century “laws of nature,” and this is
understandable too. Like in any other professional business, physics
professionals must be able to communicate with each other and they must have
a standard language that all agree upon. In this language words such as
“observer” or “law” do not have their colloquial meanings; they have
technical meanings. For a non physicist it is important to see through this
high level technical language to understand the fundamental physics content
that physicists already know.

It is always possible to put the prose words into precise mathematical
statements. So when a physicist says “Newton’s laws” he has precise
mathematical definition in mind. It does not matter if it is called a
“law, ”a definition,” whatever. As you say, physicists never confuse
notation with meaning. That’s why a non-physicist must always be clear about
the meanings.
>
> > We can write the first derivative in delta notation as the increment in
> > distance x divided by the increment in time t: delta x / delta t.
>
> Just keep in mind that this is only an approximation. The exact result
> is recovered only when delta t goes to zero.

What do you mean by exact? To me the distance the moon moves in delta t =
1/1000 seconds is an exact observation. Delta t is infinitely distant from
0, but it is exact. What I mean is that no new insight is gained by
considering the case t-->0. In practice you choose the smallest unit you can
measure.
>
> > But
> > the distance moved in the first increment of time equals the
> > acceleration. Therefore, we can write
> >
> > delta x/delta t = [-grad V].
>
> This is completely incorrect. What you should have written is
>
> delta (delta x/delta t) / delta t = -grad V.
>
What I wrote is elementary physics. I am saying that velocity in the first
unit of time equals acceleration. You are saying this is completely wrong.

> grad V is a vector attached to every point in space...

First, thanks for this explanation. Instead of considering the most general
case, let’s look at the specific case of orbits. I believe grad V could be
applied to orbital situation. There is no doubt that the method of studying
motion with the gradient vector point of view may be useful in some
situations. But in this case, it appears to me, the complications that ensue
by introducing this notation is unnecessarily high for physics returns we
get. Take the case of circular orbits. You divide the circle into points.
These points are spaced with equal intervals. You attach an arrow to each
point. Each arrow is perpendicular to the radius of the orbit. Each arrow
has the same length.... All this cluttering of the figure is done in order
to say that the orbit has uniform motion. What is gained here by talking
about grad V? It seems superfluous to me.

> Perhaps
> what you are trying to get at is that the definition of a conservative
> force is taylored for the proof conservation of energy.

Yes. Something like that. What does it mean when you assume “only
conservative forces?” As I understand it, by tradition physicists prefer to
call “acceleration” “force.” Substituting “acceleration” in “only
conservative force” I get:

“Only conservative [acceleration].”

“Only Conservative accelerations” may be physicists’ way of saying “let’s
consider motions where net acceleration is zero.” The trivial case of which
is the uniform motion. From a naïve point of view, it appears that your
derivation amounts to writing “acceleration” in two different notations,
then setting, acceleration to zero, and calling this situation “gradient
force.” I am not sure this is correct though.

> This is true,
> as the name "conservative" implies. Further justification can only go
> to experimental observation, and we do observe that energy is conserved
> in simple mechanical systems.

This seems not to be relevant to the derivation question. The question is
not about the observation of energy conservation but about the derivation of
it from the Newtonian laws of motion.

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