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

[Moses Fayngold <moshfarlan@yahoo.com>]

On Saturday, August 20, 2016 12:57 PM, John Denker <jsd@av8n.com> wrote:

> Now there's a problem.
> There is a force-ish thing "here" that we know based on the third
> law, and there is a force-ish thing "over there" that we know based
> on equilibrium arguments.

In my view, such formulation only creates a problem where it was not existing. Generally, there may be situations with two "force-ish things" one "here" and one "over there", but without equilibrium. We do not say that two distant point masses in vacuum equally attracting each other are in equilibrium. And the same about mentioning only one "force-ish thing "here" ...based on the third law". What does that mean?  
> From a physics point of view, these are two different things, i.e.
> two different /dynamical interactions/ ... but from a mathematical
> point of view they are the same vector.  
At this point, I agree with Richard Tarara (as well as with Bob Sciamanda and Philip Keller) that "...this is not the only place that physics fudges with the math without damage to the physics".  
> The math is unambiguous and uncompromising:  a vector has a >direction and a magnitude, but it does not have a location.  >Talking about a vector "here" or a vector "over there" does not make sense.  >It is a distinction without a difference.
  I think this uncompromising definition contradicts others even within Math itself. Thus, it contradicts the concept of a vector field based on the notion of a vector as a function of position. In vector field, to each point of space is attributed a specific vector "sticking out" of this point. If a vector "does not have a location", then a vector field does not have an existence. Another example - torque, requiring localized force (its application point). The same force applied to different points of an extended body, will produce different rotational effects. In this case, application point is as important characteristic of a force as its magnitude and direction. A related (albeit weaker) example - abstract definition of a by-vector in GA as an outer (or exterior) product A^B of two vectors with tale to point arrangement. If a vector does not have a location, then consistency of the concept of A^B becomes debatable. Of course, the defined arrangement can be preserved if we consider the whole bi-vector A^B as an indivisible object with undetermined location. This is surely possible, but it is not compelling argument since it forbids considering A and B as independent ingredients of A^B. The moment we consider A and B as independent entities, each one without specific location when considered separately of the other, the concept of their outer product becomes problematic. 
> Let's be clear:  It makes no sense to tell students that force is
> a vector and then turn around and ask them to distinguish a force
> "here" from a force "over there".  It is a crime against the laws
> of mathematics.

  In view of my arguments, it may sometimes be beneficial and even necessary to commit crimes against the laws of mathematics such as the one discussed here. The concept of vector field is a crime against the law depriving vector of location, and vice versa.   This does not mean that each vector must have a location. The momentum vector in QM is a non-local entity. I only want to point out that there are physical situations that imply vectors with specific location.  

Moses Fayngold,NJIT_______________________________________________
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