Virial theorem

The virial theorem states that the average kinetic energy of an object whose motion is bounded is given by

where r_i and F_i are the position and Force vectorss on the ith particle respectively.

If the force are derivable from a potential the theorem becomes,

If V is a power-law function of r,

then the virial theorem can be written as

In particular, for the further special case of inverse square law forces (i.e. n=-2), the virial theorem states that the average kinetic energy of the objects in equal to -1/2 times the average potential energy.

Since the gravitational force obeys an inverse square law relation, the virial theorem is a remarkably useful simplifying result for otherwise very complex physical systems such as solar systems or galaxies, and is also applicable to a number of other similar scenarios.

The theorem is also very useful in the theory of gases and can be used to derive Boyle's Law for perfect gases.

The virial theorem takes its name from the quantity known as the virial (Latin for "force"), defined as:

where r_i and p_i are the position and momentum vectorss of the ith particle respectively.

The virial theorem can be derived by considering the properties of the virial in the limit over a long period of time.

External links

• The virial theorem made easy by John Baez
• A mathematical analysis of the virial theorem