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The virial theorem states that the average kinetic energy of an object whose motion is bounded is given by
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where ri and Fi are the position and Force vectors on the ith particle respectively.
If the force are derivable from a potential the theorem becomes,
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If V is a power-law function of r,
- V = arn + 1
then the virial theorem can be written as
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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:
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where ri and pi are the position and momentum vectors 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.
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