Re: [Phys-l] Equations (causal relationship)

[John Denker <jsd@av8n.com>]

Brian McInnes wrote:

> > Surely you know that displacement vectors add.
> >   -- 1 meter north plus 1 meter west equals 1.414 meters northwest
>
>
> Such displacement vectors do not occur at the same time - the notion  
> is that the displacement to the north is followed by a displacement  
> to the west and the final position of the particle is away to the  
> north-west.

I'm trying to figure out where this is coming from.  I'm trying
to be sympathetic ... but really not succeeding.

I did *not* say 1 north west "then" 1 meter west.
I did *not" say that and I did not mean that.
The notion that "the displacement to the north is followed by a
displacement to the west" is entirely without foundation.

AFAICT all of the statements about the alleged cause-and-effect character
of the F=ma law are in this same category:  they are entirely without
foundation, and exist only as opinions in the mind of certain parties.

By way of example, here's a scenario:  I have a wind-up car that moves
1 meter north relative to the table-top.  Also I shove the table 1 meter
west.  Maybe both displacements happen at the same time;  maybe one happens
first;  maybe the other happens first.

The displacement vectors don't care.  The acceleration vectors don't
care, either.  The assertion that there are many forces but only one
acceleration is not a law of physics;  it exists only as an opinion
in the mind of certain parties.

This is important because it reflects the uttermost foundations of
arithmetic.  Addition is commutative!

Repeat after me:
         Addition is communtative!  ADDITION IS COMMUTATIVE!

A plus B is equal to B plus A.  When I say A plus B, I mean A plus B.
I do not mean A "then" B.

Also by way of non-sympathy, let me point out that my statement about
displacement vectors could have been seen as an integral(dt) of the
following statement about velocity vectors.

Similarly, my statement about acceleration vectors can be seen as the
time-derivative of the following statement about velocity vectors.
Since position vectors can be added, it immediately follows that
acceleration vectors can be added.  (Addition is linear, and differentiation
is linear.  Again I am appealing to mathematical principles that are
the bedrock of everything we do.)

> > Surely you know that velocity vectors add
> >   -- wind triangle:  aircraft velocity (relative to air)
> >      plus wind velocity (relative to ground)
> >      equals aircraft velocity (relative to ground)
> >      [with due regard to sign conventions]
>
>
> Yes, I do.  In your example the aircraft has has a motion which is  
> described and measured as its velocity and the value of that velocity  
> will depend on what frame of reference one wishes to measure it in -  
> with respect to the air, with respect to the ground or, if one  
> wishes, with respect to the passengers in the plane.  In each frame  
> there is only one value for the velocity

No, the velocity vectors I described are frame-independent
because they are (by construction) explicitly velocity-difference
vectors.

This is analogous to the way that the _delta_ V across a resistor
is gauge-independent, even though V itself is gauge-dependent.

The criticism of my example is entirely without merit.
It looks to me like grasping at straws.

The community has a choice:
 -- We can accommodate a few people who have strongly-held opinions, or
 -- We can rely on the axioms and bedrock principles of mathematics.

I've always had good luck going with the principles.  I recommend we go
with the principles.