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*From*: John Denker <jsd@AV8N.COM>*Date*: Tue, 3 Feb 2004 09:59:02 -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?

Previously I said I didn't know of one.

Update: Now I know there isn't one.

Consider two potentials, marked with ' and ".

' ' ' ' ' ' '

'

'

'

" " " " " " " " " " "

'

"

' <-- X

"

'

That is, at point X, the two potentials agree as to the slope

of the potential (i.e force), but one of the potentials has

only half the binding energy of the other. I repeat:

same force, different energy. This situation arises naturally

e.g. if you compare the Coulomb potential to the Yukawa potential.

It can be demonstrated in the classroom by rolling marbles in

suitable dishes.

The virial itself (p dot r) involves only local properties. Force

is a local property; depth of potential (i.e. binding energy)

is *not* a local property. There is no measurement a particle

at point X can make to ascertain which of the two possible

potentials it's in.

You might say the fundamentally the virial theorem starts by

relating the KE to the local force. From there, it exploits

the fact that for a power-law force, the slope of the potential

is proportional to the depth of the potential divided by r.

This fact is a fact, but IMHO it is little more than a mathematical

curiosity. In any case, you won't know this fact unless you

differentiate the potential or do something equivalent.

If you try to derive the virial theorem result using energy

principles alone, you're going to get the wrong answer for

at least one of the potentials diagrammed above, since they

have the same force and same KE but different binding energy.

I know Brother Blais wants to restrict attention the gravitational

1/r potential ... but the energy principles don't know that. To

make the result stick, at some point you will have to bring in

the *local* shape of the potential (not just its total depth)

and that is tantamount to bringing in the force.

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