The virial theorem relates time-averaged kinetic energy to the forces in a mechanical system. For particles with positions 
, momenta 
, and differentiable
potential energy 
,
define 
and 
.
Newton's equations give
 |
(1)
|
If the time averages exist and ![[D(tau)-D(0)]/tau->0](/images/equations/VirialTheorem/Inline6.svg)
as 
, then
 |
(2)
|
where 
denotes a time average. Bounded 
is a sufficient condition for the boundary term to vanish
(Fowler).
If 
is a homogeneous function of degree 
, so that 
for 
, differentiating with respect to 
gives 
. Consequently,
 |
(3)
|
For Newtonian gravitational attraction, 
and 
. For a quadratic potential, 
and the time-averaged kinetic and potential energies are
equal. These conclusions require the averaging and boundary-term hypotheses; they
do not apply to an arbitrary instantaneous state.
