Re: [Phys-l] bound vectors ... or not

[John Denker <jsd@av8n.com>]

On 09/07/2010 06:02 AM, Folkerts, Timothy J wrote:

> I'm saying if I tie Rope B to a Object A, the force F(on A due to B)
> *is* applied to a specific point (or can at least be integrated to one
> effective point).  

We agree that the physics of the situation requires us 
to specify the point-of-application as well as the 
force-direction-and-magnitude.

The question is:
 a) does the concept of "force" include just the vector
  describing the direction-and-magnitude of the push, or
 b) does the concept of "force" include /two/ vectors,
  i.e. both the point-of-application and the push?

This strikes me as interesting and important question.
If you had asked me a couple days ago whether I knew what
a "force" is, I would have said sure, I know what a force
is.  But now I realize that my concept of "force" has been
somewhat ambiguous all these years.

> The vector we use to describe the magnitude and direction of the force
> is not "localized' or physically bound anywhere.  That same vector could
> be used to describe the magnitude and direction of any number of other
> forces (perhaps the force of you pushing on the other side).  

Agreed.

But the question remains:  Is this vector the same as "the"
force, or it just one of the two vectors that make up "the"
force?

>   To do vector addition, I can move the vectors around wherever I
> want (but that doesn't move where the rope is tied).

Agreed.

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

Faced with an ambiguity of this sort, it may be helpful, at
least for the sake of discussion, to accept *both* possibilities
and give them distinct names.  We can always throw one of them
away later if we don't like it.

So let us at least temporarily define
 -- an unbound force Fu, and
 -- a creature X which is an ordered pair 
        X = (r, Fu)
    where r is the position vector specifying the 
    point-of-application.

Some folks would call X a "bound force" but others (including
Philip K.) would argue that this is a misnomer, i.e. that Fu
is "the" force and X is not a force.

I remark that X really is an ordered pair (r, Fu) and not
merely the wedge product (bivector) r /\ Fu  [aka the cross
product (pseudovector) r x Fu].  Given an ordered pair (r, Fu)
we can always pick out the second element Fu, whereas given
only the product r /\ Fu we cannot reconstruct r or Fu.

It appears that equations such as
   F = ma
   F = dp/dt
   F = - dE/dx
involve only Fu not X.  So as far as the mathematical formalism
goes, it appears that F (usually?) means Fu ... and that oddly
enough, the concept X has heretofore gone without a symbol.

Bob S. pointed out that X is in some ways a watered-down version
of the _vector field_ F() or a selected point F(r) in the field.
This raises pedagogical issues, because we want to explain force 
and point-of-application to students who are not yet ready to 
cope with vector fields.  The vector field concept is tried-and-
true, but it is overkill for simple point-particle mechanics
problems, in the sense that the vector field is defined for an 
infinite number of points, be we only care about the handful
of points where the particles happen to be.

The same questions apply to lots of things, not just forces.  
For each particle in a system of point-particles, and for each
parcel of fluid in a continuous system, again we have a choice:
  a) Should we talk about the position (r), the net force (F),
   the momentum (p), the angular momentum (L), the entropy (S),
   et cetera, or
  b) Should we talk about the so-called "bound force" (r,F),
   the "bound momentum" (r,p), the "bound angular momentum"
   (r,L), the "bound entropy" (r, S), et cetera?

Note that (r, S) cannot qualify as a "bound vector" since
S is a scalar.

It seems to me that the conventional approach is to describe
the parcel using one big "dynamical state variable" i.e. the
ordered tuple (r, F, p, L, S, ...).

To this way of thinking, the aforementioned "X" is seen as a
type of dynamical state variable.  If two states X1=(r1, F)
and X2=(r2, F) share the same F, we can say that the force
F is the same, but the dynamical situation is different.

I'm not saying that's the only way of thinking about it, but
at first glance it seems attractive.  It allows my head to
stop spinning, and (perhaps) allows me to talk about these
things without contradicting myself every few minutes.

To this way of thinking, we can say
 -- A so-called "bound vector" is not a vector but rather
  an ordered pair of vectors.
 -- A so-called "bound force" is not a vector and is not
  a force, but rather a dynamical state variable consisting
  of a pair of vectors (r, F).

This way of looking at things has the advantage that it
uses the word "vector" in a way that is consistent with
the axioms of linear algebra.

Comments?  Suggestions?