Re: Orbital Kinetic Energy

[John Denker <jsd@AV8N.COM>]

Quoting Brian Blais <bblais@BRYANT.EDU>:

> I was wondering if anyone knew of a calculation showing that the kinetic
> energy of an object in a circular orbit is equal to half of the potential
> energy, where the calculation does *not* use acceleration or force at all.
> Is there an argument for this based purely on energy concepts?

0) Be careful, the result as stated only applies to objects
orbiting in a 1/r potential (although generalizations are
possible to other power laws).  Not all potentials are
power-laws:  consider the Yukawa potential just for starters.
Also the result assumes the KE is PV/2, which is by no
means the general case.  Plus other caveats and restrictions
that cannot be taken for granted.

1) Certainly it is possible to derive the viral theorem
without mentioning acceleration.  The standard derivations
don't mention it.  Indeed you don't need to know the masses
of the particles involved.

Look at
  http://math.ucr.edu/home/baez/virial.html
about halfway down, in the section called "the proof".

2) The standard derivations do mention the force.  I'm
not quite ready to say this is *provably* necessary, but
speaking for myself I don't see any easy way around it.
Forsooth the name "virial" means "force" and it would be
a little bit odd if the virial theorem didn't have anything
to do with the force.

A critical step in the standard derivation involves expressing
the force as the derivative of the potential, and then
invoking the fact that (by hypothesis) we are dealing with
a power-law potential, so that multiplying by a constant
times r "mostly" undoes the derivative.  So it looks like
the force is deeply implicated.

3) Why do you care, anyway?  If you know the potential,
you implicitly know the force, and conversely if you know
the conservative force you implicitly know the potential,
plus or minus an arbitrary gauge term.

As it says in MTW, if a complicated calculation gives a
simple result, we should look for better methods.  For
sure the derivation of the virial theorem is a mess, but
I'm not convinced that a simpler calculation is possible,
mostly because I'm not convinced that the result is as simple
as the usual (mis)statements suggest.  A conclusion is very
simply stated ("half") but if you actually track down the
premises (non-relativsitic blah blah power-law blah blah
gauge blah-blah bounded-this-and-that etc.) it seems like
a miracle that the calculation holds together at all.


Tangent:  The virial theorem is occasionally useful, but
it gives me the creeps because it is not gauge-invariant.
You would think that all the physics would be the same for
either
  -- (1/r) potential
  -- (1/r + const) potential
but the virial theorem, in its usual form, only applies to
the 1/r potential.  Am I the only one bothered by this?