Re: [Phys-L] treating force as a vector ... consistently

[John Denker <jsd@av8n.com>]

On 09/13/2016 09:18 AM, David Craig wrote:

> engineers distinguish whether or not vectors are associated with a 
> point as a matter of course.

Well, some do it properly and some don't.  In any case, there is
enough confusion that we ought to be aware of it.

Let's be specific:  The usual physics approach says:
  1) The position x is a vector.
  2) The velocity is v = dx/dt.
  3) The acceleration is a = dv/dt.
  4) The force is F = ma.

That's not a complete description of the physics, but it is correct
as far as it goes.  If you try to re-interpret it in terms of bound
vectors, it's a mess.  In particular, assuming you represent a bound
vector by two points ("tip" and "tail"), what happens when you
differentiate it?  Do you differentiate the tail as well as the
tip?  It's a mess if you do and a mess if you don't.

When the same name (e.g. "vector") is used to refer to two different
concepts, you are guaranteed to have trouble.  It's worse when the
ideas are somewhat overlapping (e.g. "vector") ... as opposed to
wildly different, as in "dove" (the past-tense verb) versus "dove"
(the bird).

One can argue whether something counts as a "misconception" when it
is required in one course and forbidden in another.  Certainly it
becomes a misconception when students (or others) incautiously lift
the word from one context and use it in the other.  This is more
problematic than ordinary run-of-the-mill misconceptions.

> If I recall the terminology correctly, one of the principal statics 
> textbooks refers to vectors attached to a particular point as 
> “rooted” vectors.

"rooted vector" == "located vector" == "bound vector"

A related concept is "force plus line of action".  Yet another concept
is "force plus point of application".

The physically-sound alternative is *force* plus *torque* .  Virtually
all physics texts do it this way.  Advanced engineering texts do it
this way.

By way of analogy:  I've taught a lot of kids how to ride bicycles.
IMHO training wheels are almost always more trouble than they are
worth.  It's an idea that I keep down in the farthest corner of my
bag of tricks, for very unusual situations.  I put bound vectors
in the same category.

> As an aside, the concept John D was describing (without naming it) 
> was that of a cross section of a fiber bundle.  A vector field is a 
> cross section of a manifold’s tangent bundle.

Yes, but first things first.  When the audience doesn't have a
firm grasp of what a vector is, or how a vector field differs
from a vector space, you can't start with tangent bundles.

In fact a tangent bundle is defined in terms of tangent spaces,
so it makes sense to do tangent spaces before tangent bundles.

There is a rather general pedagogical principle that says "concept
before name".  You can teach people pretty much everything they
need to know about tangent bundles without ever using the word.

> One is free to dismiss it if you like, but when you are doing things
> like GR it really is a useful idea.

That's ambiguous, due to a dangling antecedent.  Does "it" refer
to rooted vectors, or to fiber bundles?

Let's consider both possibilities:

 -- I would not /entirely/ dismiss the idea of "force plus line
  of action", because it is sometimes useful, e.g.
      https://www.av8n.com/physics/ill-posed.htm#sec-yo-yo

  On the other hand, I would be careful not to take it as a
  starting point, because the line of action is sometimes
  undefined, undefinable, and uninteresting, e.g. for a
  pure torque.

  This stands in contrast to the concept of force plus torque,
  which is operationally simpler and 100% reliable.

  In any case, even when using the line of action, I reserve
  the word /force/ to refer to a plain old vector, with direction
  and magnitude but *not* location.  If I wish to talk about force
  and torque, or force and line of action, I will mention both
  explicitly.

 -- As a separate matter, I would not dismiss the idea of fiber
  bundle.  I just wouldn't bother with it on the first trip
  around the spiral.  Those who need it can pick it up on a
  later trip.
     https://www.av8n.com/physics/spiral-approach.htm