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*From*: John Denker <jsd@AV8N.COM>*Date*: Mon, 2 Feb 2004 23:38:28 -0800

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?

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