Virial Theorem Website Links For
Theorem 		 

Information About

Virial Theorem
  CATEGORIES ABOUT VIRIAL THEOREM
   
classical mechanics
   
physics theorems
   
fundamental physics concepts
   
SHOPPER'S DELIGHT
 
Shop For:  
Department
Price Minimum:
Price Maximum:
      



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

where r''i'' and '''F'''''i'' are the Position and Force Vector s on the ''i'' th particle respectively. The bar over top of an expression represents time-averaging of the overlined quantity for the entire system.

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

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

If ''V'' is a power-law function of ''r'',

: V= a r^{n+1}

or if ''V'' is the sum of terms ''V''''ij''(''r''''ij'') of this form, then the virial theorem can be written as

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

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


Equivalently:

  • 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 System s 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 (rooted in the Latin vires, "forces"), defined as:

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

where r''i'' and '''p'''''i'' are the Position and Momentum Vector s of the ''i''th particle respectively.

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

In quantum mechanics, the Virial theorem can be written as :
2\langle K angle = \langle r \cdot
abla V angle


UNBOUND CASE

Without assuming boundedness we have the following more general properties for gravitational systems:
  • the time-average of the potential energy of the system is equal to twice the total energy minus the average time-rate of change of ''G''.

  • the time-average of the kinetic energy of the system is equal to minus the total energy plus the average time-rate of change of ''G''.


For two simple cases:
  • For a parabolic orbit the properties still apply: the averages are zero, ''G'' increases slower than ''t''.

  • For a hyperbolic orbit the time-average of the kinetic energy is equal to the total energy, without the minus, the average rate of change of ''G'' is the difference between the value and minus the value, that is twice this value.



EXTERNAL LINKS