Re: visualizing a non-potential

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

On 04/26/2003 11:39 PM, Bob LaMontagne wrote:
> > The force vector acting on the electron has components
> > proportional to
> >        Fx =  y/sqrt(x^2+y^2)
> >        Fy = -x/sqrt(x^2+y^2)
> >        Fz =  0
> >
> > Note that the magnitude of F is constant everywhere,
> > and that F points clockwise everywhere.
...
> integrating the dot product of F and ds around a complete circle of unit
> radius in the xy plane centered about the origin gives 2 PI - going around
> twice gives 4 Pi, etc. - definitely not a potential.

:-)

> But your depiction always comes back to the same value regardless of how many
> times you pass through the same point.

Again, it is essential to be clear about what
function is being discussed.

-- The _energy_ is not a function of position.
  Returning to the same position does not
  return you to the same energy.  The diagram is
  not intended to portray height above sea-level.
  Any appearance of definite height is a bug.

++ In contrast, the _force_ is a function of position.
  The diagram is intended to depict a definite slope:
  clockwise = locally downhill everywhere.

> What I was curious about is how one
> creates a function which seems to constantly increase along a closed path but
> gives the same value when one returns to any point on the closed path.

Ditto.

The energy constantly increases along certain closed paths.

The force returns to the same value when one returns to
any given point.  The energy does not.