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Re: [Phys-L] point particles puzzle



the system is not reductible in terms of particles

there are particles and microstrings (elastic force)

the universe have not only particles, also have forces



2014-06-30 19:09 GMT-03:00, John Denker <jsd@av8n.com>:
On 06/30/2014 11:17 AM, Carl Mungan wrote:

If one subdivides the spring into portions [...]
then one end moves farther than the other end
and so the equal and opposite forces at their two ends do not do equal and
opposite work, and each portion of the spring thus gains a portion of the
elastic PE.

Exactly so.

In my original note I used the word "parcels"
instead of "portions", but the idea is the same.
The portion-size can be small, but it cannot be
zero. The portions cannot be pointlike in the
usual sense. They have to be stretchable.

1) There is some nontrivial physics here: By
making the parcels small enough, we can make
their /spin/ angular momentum negligible compared
to their /orbital/ angular momentum.
-- The spin angular momentum of each parcel is
computed relative to the center of the parcel
itself, whereas
-- The orbital angular momentum is computed
relative to some chosen datum, some fixed axis.

To summarize, AFAICT the usual wording of the laws
of motion in terms of point particles is open to
misinterpretation, to say the least; for details
see item (3) below. On the other hand, it is
easily fixable. The idea is to think in terms of
small-sized parcels, rather than zero-sized points.

The idea of having a huge number of subsystems aka
parcels aka portions is routine in many branches of
science and engineering. Finite-element modeling
is used in fluid dynamics, stress/strain analysis
of materials, the design of magnets and other
electromagnetic devices, et cetera. OTOH it usually
isn't mentioned in the introductory physics course.
The tendency is to consider the spring (or whatever)
as a collection of zero-sized points ... if its
internals are considered at all.

As a separate matter, non-experts tend to
divide the world into "the" system and "the"
environment, where the system and environment
are treated very differently. Sometimes you
can get away with that, but I don't recommend
it. Often it is preferable (or necessary!)
to divide the system into a huge number of
subsystems, each of which is treated on the
same footing, as in the spring example we
have been discussing.

=======

2) Here is another way of making the problem go
away. This way is nice and practical, recommended
for a wide range of applications. It is suitable
for students (and others) who are not comfortable
thinking about infinitesimals.

We allow the laws of motions to apply to rigid
bodies, even ones with a nontrivial non-infinitesimal
size. To make this work, we have to declare that
the force laws (in their usual form) are incomplete;
not wrong, just incomplete. We need to supplement
them so that we keep track of the /torques/ as well
as the forces. For each force, we need to keep
track of the point of application (or at least the
lever-arm), not just the direction and magnitude
of the force itself. We need to keep track of
the /angular momentum/ as well as the ordinary
linear momentum.

This is a big win for finite element modeling,
because it means that the elements don't have to
be infinitesimal. A modest number of /moderately/
small elements is vastly more efficient than a
super-huge number of infinitesimal elements.

========

3) Here is yet is another way to make the problem
go away. It allows us to salvage the formulation
in terms of "points".

From a sufficiently sophisticated point of view,
it could be argued that we can have zero-sized
points and still allow them to be stretchable.

Consider the Dirac delta distribution δ(x) and the
stretched version δ(x/k) for some nonzero stretch-
factor k. Both distributions are pointlike in the
sense that they have zero width in the x-direction
... but they are not equivalent. Specifically,
∫ δ(x/k) dx = k ∫ δ(x) dx [1]
as you can trivially verify by a change of variable.

So, sometimes a thing with zero width is stretchable.
OTOH in the algebra-based physics course, or indeed
in the calculus-corequisite physics course, you don't
want to go anywhere near equation [1]. Students
struggle with that idea even in the junior-level
course on the mathematical methods of physics. At
the introductory level, and several levels beyond
that, the idea of stretching a zero-sized point is
counterintuitive.

By way of contrast, note that the step function
H(x) is invariant with respect to stretching;
H(x) = H(x/k). The δ distribution is the
derivative of H, which gives you another way
of deriving equation [1].

From the sophisticated point of view, expressing the
laws of motion in terms of "points" is not wrong.
However, given that few if any of the students are
operating at this level of sophistication, the
"point" formulation is open to misinterpretation,
to say the least.

Indeed, the idea that the points themselves "must"
get stretched cannot possibly be intuitive at any
level, because it is not always true. By way of
counterexample, when the universe expands, the
galaxies do /not/ get larger; they just get
farther apart.
http://www.av8n.com/physics/expansion-of-the-universe.htm

So when formulating the laws of motion, the usual
formulation in terms of point particles is at best
sloppy and/or incomplete. To clean it up would
require a detailed explanation as to what is
stretchable and what is not.

Tangential technicality:

Physicists invented δ(x) and used it for many
decades before mathematicians found a way to
formalize it, to make sense of it in systematic
mathematical terms.

It took people a good long while to realize
that even though physicists were in the habit
of calling it a delta function, it's not really
a function. It does not uphold the mathematical
definition of function. Instead it should be
called a /distribution/. The word "distribution"
doesn't tell you much, because it is a term the
mathematicians made up specifically for this
purpose ... but I reckon an uninformative name
is better than an invalid, misleading name.

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--
Diego Saravia
Diego.Saravia@gmail.com
NO FUNCIONA->dsa@unsa.edu.ar