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

[John Denker <jsd@av8n.com>]

On 12/11/2008 08:35 AM, Savinainen Antti wrote:

> Modern classical mechanics defines the first law with reference to an
> inertial system or inertial reference frame. 

The same can be said for all the laws, in the following sense:

In general, there are some "prerequisites".  For the first law,
we need notions of reference frame, position, velocity, and
maybe acceleration (or lack thereof).  For the second law, we 
need all that plus mass, force, and nonzero acceleration.

So it seems to me we need to decide how to handle these
prerequisites.  If we arbitrarily lump all the prerequisites
in with the first law, then obviously that makes the first
law indispensable.  However, I don't recommend this "lumping" 
approach, because it departs dramatically from the normal, 
conventional statement of the first law.

I would argue that one should instead state the prerequisites
separately, in which case the first law becomes, once again,
merely a corollary of the second law.  (I suppose one could lump
the prerequisites in with the second law, but I don't recommend
that either.)

> > From the point of view of teaching *early* introductory physics I
> > don't mind making Newton's first law as a corollary of the second
> > law.

I'd turn that around 180 degrees.  From an advanced point of
view, I consider the first law to be a corollary of the 
second law.  I assume there is an operational way of measuring
force.  Arrange for the force to be zero, apply the second
law, and integrate both sides.  Use this to define "inertial
frame" if necessary.  Verify that what's true in one inertial
frame is true in another (empirically and/or by differentiating
the integral we just did).

Meanwhile, in an introductory course, I don't mind taking 
things one step at a time, introducing the less-general
result (the first law) before moving on to the more-general
result (the second law).  The conceptual hurdle to understanding
the first law is pretty tall for anybody who is seeing it for
the first time.

And the first law is not wrong.  It doesn't need to be unlearned.

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

I consider these questions of what is derivable from what, and
what is a corollary of what, to be interesting but of mostly
_secondary_ importance.  I care more about what's true than
about what is the "best" way to derive it.  Usually there's
more than one way to derive it.

In contrast, I am much more exercised about the distinction
between a force (narrowly speaking) which does not have an
attachment point, and whatever the other thing is that has
both a force and an attachment point.

Either a force is a vector or it isn't.  When we draw diagrams,
either the attachment point of the little arrows matters or it
doesn't.  It seems to me we face the risk risk of subtle but 
pernicious misconceptions creeping in.

Here's a half-baked thought.  Maybe when we introduce the
idea of attachment points, we should choose an origin and
actually draw the position vector as well as the force
vector, and formalize the torque as

     torque = arm /\ force

     BTW, in a previous note I think I wrote force/\arm 
     which is backwards i.e. unconventional and not
     recommended.  Sorry.

On the second say maybe draw the position vector as a
dashed line.  On the third day draw it as a dotted line.
On the fourth day omit it entirely, saying it is implicit
even when it is not drawn explicitly.

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

The same problems (or sometimes deeper problems) arise
whenever we draw a vector _field_ e.g. an electric field 
or the velocity field in a fluid.  We are overloading 
the little arrow symbol to represent _two_ vectors:
 *) the direction and magnitude of some vector, and
 *) the position vector that tells where the other
  vector is located.

This is a tricky business.  The ideas are not particularly
deep, but it seems we don't have a good language for
talking about the ideas.  Are the following things "the
same" or not?

      (A) ------>


                  (B) ------>

As vectors, they represent the same vector, i.e. the same
direction and magnitude.  But there's more to physics
than vectors.  And if those things are not vectors, what
are they?  Previously I mentioned possibly formalizing
things in terms of bivectors, which is fine for angular
momentum but not adequate for electric fields, where
we care about ∇ • E not just ∇ /\ E.

The mathematics of vector fields is not in doubt.  The
equations are fine.  It just seems that our pictures and 
our prose are not always sufficiently expressive.