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Re: [Phys-l] basic laws of motion +- vectors +- angular momentum



Thanks very much for bringing this up again. It has bugged me for ages.

Is there any way to "work backwards" from quantum mechanical angular
momentum ideas to some sort of classical analog?

And is rotation a relative or absolute concept? I have no problem with
imagining a particle moving to the right in a 'stationary' universe as being
the same as the entire rest of the universe moving to the left while the
particle is stationary, but I really have a big problem with imagining the
whole universe rotating around a particle instead of the particle spinning
while the universe just sits there. Do I just not have a good enough
imagination?

Steve Highland
Duluth MN


Hi Folks --

I'm conflicted about the following:

1) As I see it, Newton's third law implies conservation of
momentum ... and vice versa. They're equivalent. So far
so good.

Conservation of momentum means conservation of *linear*
momentum.

2) Conservation of angular momentum implies conservation of
linear momentum ... but not vice versa. Hint: consider
angular momentum about an axis infinitely far away.

3) Combining (1) and (2), I don't see any way, starting
from the usual statement of Newton's laws, to derive
conservation of angular momentum ... not without
additional assumptions.

4a) In a previous discussion of this topic, somebody said
we should start with point particles, and assume all
forces are central forces. Then larger objects can be
built out of point particles plus forces of constraint.
I guess that would be logical, but I'm not convinced it
is physically correct. For example, consider the
magnetic force between two current-carrying wires, or
the force on an electron in a betatron, or anything
involving the Poynting momentum. It's not at all
obvious that these forces are pointlike or central.

4b) An alternative would be to just introduce conservation
of angular momentum as a fourth law of motion. The third
law would retain its usual form and apply only to linear
momentum. As such, it would be demoted to a mere
corollary of the fourth law ... in much the same way as
the first law is merely a corollary of the second law.

I'm happy to retain the first law and the third law
because of their historical importance and pedagogical
usefulness ... but that doesn't prevent us from
superseding them with more powerful and systematic
laws (the second and fourth).

4c) Of course if we were serious about formalizing
things properly, we wouldn't start with Newton's
laws. We would start by writing down a Lagrangian.
That would tell us what's conserved and what's not.
Excellent reference:
http://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/index.html
http://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/book-Z-H-3.html

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

Here's another way of looking at the whole issue. This
may explain why it is hard to think clearly about this
topic.

We need to carefully distinguish
-- natural, physical reality
-- our intuition
-- our formal models and equations

Just because something is true physics-wise doesn't mean
it is logically derivable formalism-wise. Gödel had
something to say about this.

A) In this case, we formalize force as a vector. Formally,
vectors have direction and magnitude, period.

B) Meanwhile, I suspect that many of us, and many students,
have an intuitive force-like notion that involves direction,
magnitude, and *point of attachment*.

There is a huuuge difference between (A) and (B), as we
can see from the following diagrams:

<----------
action (A)

---------->
reaction (A)

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


<---------- ---------->
action (B) reaction (B)


You can see that diagram (A) conserves momentum and
upholds the usual vector-based version of Newton's
third law, but violates conservation of angular momentum.

Meanwhile, diagram (B) upholds conservation of angular
momentum as well as conservation of plain old linear
momentum.

The point is that in terms of vectors, strictly speaking,
there is no difference between diagram (A) and diagram (B).
Vectors have magnitude and direction, period.

If you want to talk about something having magnitude,
direction, and point of attachment, you need a bivector:

torque = force /\ lever_arm

So ... as far as I can see, so long as the third law is
formalized in terms of forces i.e. vectors, the formal
law does not capture the full physical reality.

Or am I missing something?

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