Re: position vs displacement

[John Denker <jsd@AV8N.COM>]

Hugh Logan wrote in part:

>>If
>>forced to think about it,  I always thought of a position vector as a
>>displacement from the origin, but it seemed inconvenient to think of a
>>displacement as a change of displacement, although that is what it
>>really is.

Larry Smith wrote:

> Randy Knight's new book (PSE, 1e) is the first that takes a consistent
> approach, IMHO.  He does as you, Hugh, suggest: x is position and \Delta x
> is displacement.  At least this year I have a consistent story for my
> students rather than apologizing for the author's own confusion in previous
> texts.  But I don't know if I'm 100% totally happy yet, because I also
> agree with the strangeness indicated in your last paragraph.

I predict people will end up 100% happy with the idea that a
displacement is a difference between two position vectors.

The physics of the situation is AFAICT the physics of pre-Galilean
relativity of position:  Each observer is free to choose to choose
their own origin of coordinates for expressing positions.  Note that
this is relativity of *position*, quite distinct from the relativity
of velocity à la Galileo and Einstein.

Therefore a position such as A or B or C is expressed as a gauge-
dependent (i.e. origin-dependent) vector.  However the principle
of relativity of position tells us that the laws of physics never
depend on such gauge-dependent vectors.  All the laws of physics
instead involve gauge-independent expressions such as A-C (the
displacement from C to A) or B-C (the displacement from C to B).

In my book:
         "displacement"     is synonymous with
         "relative position".

It is common but not really necessary for displacements to
be measured relative to some *fixed* reference. That is, the
expression A-C sometimes implies that C has zero velocity or
at least zero acceleration ... but this is not required.
Gauge-independence is what really matters;  the rest is fluff.

At the drop of a hat, any position can be turned into a
displacement (or vice versa) by choosing some point O as the
origin.  The position A corresponds to the displacement A-O
and vice versa.

The definitive check as to whether something expresses a
displacement or not is to subtract O from each of vectors
in the expression, and see if O drops out.
   A - B --> (A-O) - (B-O) = A - B (independent of O)

Note that the difference between two displacements is also a
gauge-invariant expression ... so I am happy to call it a
displacement as well.

> Furthermore, while there is a name for \Delta x (i.e. displacement), there
> isn't a corresponding one-word name for \Delta v.

Can't help you there.  The corresponding idea is relative velocity,
and I just call it relative velocity.  Not a big problem.

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

Pedagogical remark:  The idea of relativity of position is obvious
to us.  It's not a big surprise or a burden on students, but it
needs to be discussed explicitly.  It gets short shrift in a lot
of texts.