Virial theorem

The virial theorem states that the average kinetic energy of a system of particles whose motions are bounded is given by

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: overline{K} = -rac{1}{2} overline{sum_i mathbf{F}_i cdot mathbf{r}_i}

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where ri and Fi are the position and force vectors on the i th particle respectively.

Related Topics:
Position - Force - Vector

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

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: overline{K} = rac{1}{2} overline{sum_i abla mathbf{V} cdot mathbf{r}_i} .

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If V is a power-law function of r,

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: V= a r^{n+1}

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then the virial theorem can be written as

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: overline{K} = rac{n+1}{2} overline{V} .

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In particular, for the further special case of inverse square law forces (i.e. n=-2), the virial theorem states:

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• the time-average of the kinetic energy of the system is equal to -1/2 times the time-average of the potential energy

Equivalently:

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• the time-average of the potential energy of the system is equal to twice the total energy
• the time-average of the kinetic energy of the system is equal to minus the total 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.

Related Topics:
Solar system - Galaxies

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

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Note that e.g. in the case of a solid or liquid celestial body, there are gravitational as well as reaction forces, so the potential of the total force does not satisfy a power-law. In the case of elastic collisions the reaction forces act only a short time and the result is not affected.

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The virial theorem takes its name from the quantity known as the virial (rooted in the Latin vires, "forces"), defined as:

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:G = sum_i mathbf{r}_i cdot mathbf{p}_i

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where ri and pi are the position and momentum vectors of the ith particle respectively.

Related Topics:
Position - Momentum - Vector

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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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Virial theorem: Unbound case ►