[Phys-l] non-conservative --> non-grady ???

[John Denker <jsd@av8n.com>]

On 02/19/2008 03:01 PM, Jeff Weitz wrote:

> > Once you adopt the work-in-the-past idea, can't you drop the restriction
> > about "conservative" forces?
> >
> > There's lots of non-conservative force fields in the world.  Don't they
> > do work, too?  
> >
> >    In particular, the "electric power" sold by the electric company is
> >    almost always traceable to non-conservative fields in a dynamo
> >    somewhere.

> It seems to depend on how you define "system."  If a "non-conservative"
> force does work on a system then the sum of KE and PE of the system will
> change.  Isn't that why we call a force "non-conservative" in the first
> place, though we don't define "non-conservative" this way?

1) I agree we don't define "non-conservative" this way.

2) You can change the sum of KE and PE of a system by doing work
 with a conservative force *or* a non-conservative force ... so
 this will never be part of the definition.  

I conjecture that the discussion may be suffering from the usual
three-way confusion:
 a) conserved, which is not the same as
 b) constant, which is not the same as
 c) "non-conservative" force field.

Let's start by dealing with item (c).  The term "non-conservative"
is conventionally applied to a force field that is not the gradient
of any potential.  Example of the terminology:
  http://farside.ph.utexas.edu/teaching/301/lectures/node59.html
The field in a betatron is a familiar example of such a field:
  http://www.av8n.com/physics/non-conservative.htm

This does *not* mean that a betatron fails to uphold conservation
of energy (or conservation of anything else).  Not at all.  IMHO
the term "non-conservative" is just begging to be misunderstood.

Constructive suggestion:  I avoid using the terms "conservative"
and "non-conservative" in this context, and use the terms /grady/
and /non-grady/ instead.  A non-grady field is not the gradient
of any potential.

  Tangential but important remark:  This idea extends to things
  other than forces.  In thermodynamics we have quantities such
  as PdV and TdS that are not the gradient of any thermodynamic
  potential.  Conventional terminology calls these "inexact
  differentials" which is IMHO just begging to be misunderstood.
  First of all, "inexact" begs to be considered a synonym for
  inaccurate or imprecise, which is not what it means in this
  context, and secondly an "inexact differential" is not really
  a differential at all.

  Constructive suggestion:  The term /grady/ solves this problem
  as well.  dV and dS are grady, while PdV and TdS are non-grady.

This concludes my remarks about item (c).

As for the distinctions between conservation and constancy, the
only cure AFAICT is to use the words carefully, teach students
to use them carefully, and constantly remind them of the
distinction:

  A conserved quantity is constant *except* insofar as it
  flows across the boundary.

For gory details on this, see e.g.
  http://www.av8n.com/physics/conservative-flow.htm