From: terry@asl.dl.nec.com
Subject: A Phase Space "Slide Rule"
Originator: terry@aslws01
Organization: NEC America, Inc Irving TX
Date: Tue, 10 Nov 1992 16:05:10 GMT

Hi ya'll,
                              | p                                | p
1) Let x be a dimension       |        2) "Attach" the           *  
of space, and let the         |    x   ends of a fixed-          |\   x
vertical axis be its      ----+----    length line to the    ----+-*--
conjugate dimension p:        |        p and x coordinates:      |
                              |                                  |
                              |                                  |

3) Next, rotate the line (say countercloclwise) at about its center, letting
the ends to slide back and forth along the x and p axes:

      | p        | p        | p        | p        | p        | p        | p  
      *          |          |          |          |          |          *    
      |\  x      |   x      |   x      |   x      |   x      |   x     /|   x
  ----+-*--  ----*==*-  ----+-*--  ----*----  --*-+----  -*==*----  --*-+----
      |          |          |/        ||         \|          |          |    
      |          |          *         ||          *          |          |    
      |          |          |          *          |          |          |    

The motion along the x axis nicely summarizes the phase space relationships
of momentum and position for a particle bound in one dimension [1].  E.g.,
when x is 0, p will be at a minimum or a maximum (and vice-versa).  If you
select an x (or p), the construct immediately provides the valid conjugate
p (or x) value(s) in slide-rule fashion.

Can more complex phase space models be done in this way?  Well, at least for
1D you can raise the p "track" in the z dimension to represent potentials.

How about Hilbert spaces -- can the model be extended to an infinite number
of dimensions?  Since the conjugate coorinate pairs (xi,pi) are independent
of each other, the simplest extension to Hilbert spaces is simply to have
an infinite number of *separate* sets of fixed lines and (xi,pi) axes pairs.
Accurate, but not terribly interesting.

A less obvious question is whether an object N-1 dimensionality can in
general be made to "float" when fixed by similar rules into a N dimensional
space.  For the most obvious extension approach of, say, attaching the
vertices of a triangle to (x,y,z) axis, this does *not* work -- the triangle
will be immobile.  (One may plausibly presume this to be true for N>3, too.)

However, you can make the extension if you allow the "tracks" to be of the
same N-1 dimensionality as the objects.  E.g., a triangle in 3D space can
be made to "float" if its vertices are attached not to the (x,y,z) axis,
but to the (xy,xz,yz) planes.  An extension of the single-rigid-object model
into Hilbert space thus appears plausible.

				Cheers,
				Terry

[1] Powel & Crasemann, _Quantum Mechanics_, Addison-Weseley 1961, pp. 21-32.

P.S. -- Well, I DID want my latest generalization problem to be harder than
	the last one.  Ozark Do-Nothings (their other name, depends where you
	buy them) are the simplest members of an infinitely large class of
	similar (2D) devices.  The other devices are not very intuitive, and
	have nothing in particular to do with any of the above discussion,
	but they work quite nicely and are a bit bizarre to watch in action.

	The usual reaction when I show folks a diagram of such a device is
	is "That CAN'T work!"  Wasn't able to convince the head of the NSF
	computer science division until a friend programmed a simulation to
	animate the motion of such devices.  Seeing *is* believing...

        I am not aware of any mathematical formalisms that would be of much
        help on figuring out the general class of devices, although a knack
        for topological manipulations would probably help.  I'll give it two
        or three more days, but I'm now doubtful.  Any good topologists out
        there on this particular group?

+--------------------------------------------+-------------------------------+
| Terry Bollinger                            | Phone: 214-518-3538           |
| Advanced Switching Laboratory, NEC America |   Fax: 214-518-3499           |
| 1525 Walnut Hill Lane, Irving, Texas 75038 | Email: terry@asl.dl.nec.com   |
+--------------------------------------------+-------------------------------+