Re: [Phys-l] accelerated frames

[John Denker <jsd@av8n.com>]

On 10/27/2006 10:19 AM, Bob Sciamanda wrote:

> It may be useful to recall that kinematics from a wide class of 
> "accelerated frames" is usefully analyzed into two different effects:
>
> 1) effects that arise from referring kinematical quantities to an 
> ACCELERATING ORIGIN, and
> 2) effects that arise from referring kinematical quantities to a ROTATING 
> COORDINATE SYSTEM anchored to the above accelerating origin.

It's not obvious to me why it is "useful" to emphasize this
distinction, especially since a straight-line acceleration
can be considered a rotation about an infinitely-distant
center (infinitesimal rotation rate, infinite lever arm)
and as a corollary, acceleration+rotation can be considered
just another rotation (with some new rate and new center).

I can see how it is sometimes useful to use different language
in the different limiting cases, but I don't see how it is
useful to combine the two languages.

> R''  =  r'' + Ao + 2w x  r' + w' x  r + w x (w x r)        [1]
>
> The notation should be apparent - consult Fowles' text for details (or ask).

I'll ask.

I understand that if we define
  R := r + C + V t + Ao t^2                   [2]

then we get
  R'' = r'' + Ao                              [3]
which is
 -- independent of C (translational invariance) and
 -- independent ov V (Galilean relativity).

I also understand that if we redefine
  R := rotation(w) r                           [4]

then we get
  R'' = r'' + 2w x r' + w x (w x r) + w' x r   [5]
             Coriolis  centrifugal

which does not exhibit translational invariance, because
[4] and hence [5] assume that the rotation is a rotation
_about the origin_.

I'm puzzled about how to combine [3] and [5] to obtain [1].
In particular, were we supposed to accelerate and then rotate,
or rotate and then accelerate?  They don't generally commute.

Why/when is this useful?