Consider a set of n point masses, with the ith mass having position vector ri and mass mi. Then the total scalar moment of inertia of this collection about our origin is
now, using the fact that , we see that, assuming the masses are constant, the time derivative
where pi=mivi is the momentum of the ith mass. Let us now define a quantity equal to half of the above:
The quantity G is known as the virial.
Let us now take the time derivative of the virial:
where T is the total kinetic energy of the masses, and Fi is the force on the ith mass.
Now, let us denote by Fij the force exerted on mass i by mass j. If we have no external forces, then . Note also that for all i and j. (For i≠j, this is Newton’s third law of motion. For i=j, this says Fii=0; no object exerts a force on itself.)
Then we see that
Now, suppose that the system is a stable, bound system. Then, we expect the virial to be bounded to some finite range; thus, when one takes the time average of the derivative, , one should expect it to go to zero when the averaging is over a sufficiently long time scale; thus, since , for such a system, we get
. This is known as the virial theorem.
Now, suppose that the force between the masses is a central force; the force between two masses depends only upon the distance between them, and is directed along the line between them. Then we have a potential V(r), a function of the distance only, of which the force is the negative gradient .
Then, the force on mass i due to mass j is given by
, where is the distance between mass i and mass j.
and the virial theorem becomes
Now, if the potential is given by a power law, , then , and . Therefore, in the power law case, , and so the virial theorem becomes
For systems bound by gravity (or for electrostatic systems), we have an inverse square force, so k=-1, and we obtain .
It was in applying the virial theorem to the Coma galaxy cluster that astronomer Fritz Zwicky inferred the existence of what we now call dark matter.