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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).
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.
about halfway down, in the section called "the proof".
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.