Re: visualizing a non-potential

["John S. Denker" <jsd@MONMOUTH.COM>]

On 04/27/2003 02:31 PM, Bob Sciamanda wrote:
> Consider how one would numerically generate equipotentials, given the
> conservative vector field E(x,y):
> 1) Beginning at some chosen point (x,y), proceed an amount ds in a
> direction perpendicular to the direction of the local E field.
> 2) Continue in this manner, tracing out your path in x,y space.
> 3) This generates an equipotential line.
> 4) Choose a new starting point and repeat this procedure to generate a
> different equipotential, etc.

That's all true.

As added feature, you can cross out the word
"perpendicular" and it all remains true.  This
means that in situations (e.g. thermodynamics)
where you lack a metric, and therefore have neither
a notion of distance nor a notion of angle, you
can _still_ construct the equipotentials for a
one-form, provided the one-form is exact.
The procedure is simple:  from any given point,
shoot out trial vectors in all directions.  Make
a note of the directions in which the vectors,
contracted with the one-form, give zero.  Take a
step in each such direction, and iterate.  This
maps out the equipotential.

> Now, one can also perform this procedure using a NON-conservative field
> E(x,y).  The important difference is that the paths so generated are not
> equipotentials (No potential function exists).  I would not even call them
> pseudo-equipotentials.  They are, by construction, paths along which a
> charge can move without exchanging any energy with the E field.  They
> might be called paths of constant energy, although the value of that
> energy (for a given path) is not unique, even as measured from some chosen
> reference point.  A path may not be labeled with an energy value ( as an
> equipotential can), the energy value depends on the detailed history of
> how you got to this path, from the reference point.  This seems to be as
> close as one might get to an analogy of the equipotential concept of a
> conservative E field.

Again, all true.  True with or without a metric.

> For John's betatron field these paths of constant energy would be radial
> lines.  They may serve some useful pictorial purpose, but I fear they
> invite easy misinterpretation.

True again.

Such pseudo-contour lines are the standard method
for representing one-forms.  Standard, but problematic.

I have used this method in the past, e.g.
  http://www.monmouth.com/~jsd/physics/thermo-forms.htm#fig-star-form
but there were lots of problems.  This motivated
me to search for something better.

1) One problem is that contour lines (even for an
exact one-form) are not very easy to interpret.
They don't "grab" the viewer.  To create the visual
appearance of relief, cartographers must augment
contour lines with various additional tricks:

http://www.google.com/search?q=hillshading+hachures+illuminated

Using just plain contour lines, without some sort
of added cues about the relief, you can't distinguish
an upslope from a downslope.  You can't distinguish
a peak from a pit.

Even for exact one-forms, unadorned contour lines are
open to misinterpretation.  For non-exact one-forms,
the situation is even worse.

2) A more profound problem arises from the following
objectives:
 a) You want the contour (or pseudo-contour) line
to be a locus of constant energy.  The direction
across the lines represents the orientation of
the one-form.
 b) You want to choose the spacing between lines
so that the closeness of the lines represents the
magnitude of the one-form.
 c) You want the lines to be continuous.

For an exact one-form, you can satisfy all the
objectives by drawing equipotentials, equally
spaced in energy.

However, for a non-exact one-form, there is a
dilemma.  If you start pseudo-contour lines in
chosen places (no matter how well-chosen) and
extend them in the constant-energy directions, the
resulting lines will not have the correct spacing
to represent the magnitude of the one-form.

You will have to start new lines here and there,
in a never-entirely-satisfactory attempt to maintain
the correct spacing.

===

I made a number of valiant attempts to generalize
standard cartographic methods so they could handle
non-exact one-forms.  Time after time, the results
were extremely unsatisfactory.  I built up quite
a large catalog of things that can go wrong.
Scrutiny of this catalog, plus much head-scratching,
led to the invention of the fish-scale representation.

http://www.monmouth.com/~jsd/physics/non-potential.htm