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I've been investigating the Virial Theorem, and I've found different conditions, some more restrictive than others, for it to be applicable to a given system. According to my professor, said theorem can be applied in systems which fulfil all of these simultaneously:

  1. There is a large number of point-like masses
  2. These masses interact with one another exclusively through gravitational interactions
  3. The system is in thermal equilibrium

Nonetheless, in other texts I've found the Virial Theorem can only be applied to systems which are in hydrostatic (not thermal) equilibrium. Furthermore, condition nº $1$ doesn't hold either because I find the theorem ($2K+U=0$) also applies to a single body orbiting another one, since $E_{total}=\frac{1}{2}U$

What are the exact conditions a system must fulfil in order for the Virial Theorem to be applicable?

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I prefer to think of the Virial Theorem as an umbrella name for a set of related concepts, related in the sense that they all have one particular element in common.

The discussion of the Virial theorem by John Baez is a good reference. (My presentation here proceeds in a different order than the presentation by John Baez.)


Historically the name 'virial' was introduced by Clausius, in the context of statistical mechanics. I surmise that over time physicists noticed the same element in other contexts, so the scope of what got to be referred as instance of virial theorem widened.


The common element, as I see it, is interconversion rate of potential and kinetic energy.

I will work bottom-up; first simplest cases, and from there in ascending complexity.

I start with the two force laws that have the property that they give rise to closed orbits:
-Hooke's law
-inverse square law (such as gravity)

Let an object be circumnavigating, with the centripetal force according to Hooke's law.

In circular circumnavitating motion, with the centripetal acceleration according to Hooke's law the ratio of potential energy to kinetic energy is 1:1

That ratio is independent of radial distance; when you give the object additional tangential velocity the potential energy and kinetic energy of the object increase in that 1:1 ratio.

When you make that circumnavigating motion elongated then along that trajectory the values of the energies oscillate. Theorem: the averaged energies (over the full period of circumnavigation) are still in that 1:1 ratio that typifies Hooke's law orbiting motion.


Let an object be circumnavigating, with the centripetal force according to inverse square law.

With an inverse square law: when you shift an object from one circular orbit to another circular orbit the kinetic energy and potential energy do not co-change, instead they counterchange.

When you boost an orbiting spacecraft to a higher orbit the potential energy increases in two ways: the energy from the boost, and decrease of kinetic energy, because at higher altitude the kinetic energy of orbital motion is lower.

Conversely, when you apply reverse thrust to slow down an orbiting spacecraft the total energy of the spacecraft's orbital motion decreases, but the orbital kinetic energy goes up.

On Kevin Brown's website, in the article the virial theorem

[...] for any bound system of particles interacting by means of an inverse-square force, the average (negative) potential energy is twice the average kinetic energy.

In the case of a highly eccentric orbit: average over a period of revolution. As I understand it: with eccentric orbit: the average kinetic energy and average potential energy are in the same ratio to each other as in the case of circular orbit with the same total orbital energy.


Generalization to any force law

In general a force law will be a function of the radial distance $r$ in accordance with the following pattern:

$$ ... , \ r^{-3}, \ r^{-2}, \ r^{-1}, \ \text{constant force}, \ r^1, \ r^2, \ ... $$

[LATER EDIT]

In the previous version of this answer the mathematical statement with the label '(1)', was incorrect, and the correct statements (2) and (3) did not follow from it.

We have: for $n$ an integer, with the force proportional to $r^n$:

$$ 2 T = (n + 1) V \tag{1} $$

Subject to the condition that the applied force must accomodate the inertial mass:

In the case of Hooke's law (1) gives the relation:

$$ T = V \tag{2} $$

And in the case of an inverse square force law that gives:

$$ 2T = - V \tag{3} $$



An issue that I should have covered when I first posted this answer: how does it come about that in (1) neither the left hand side nor the right hand side has a factor $r$ for radial distance?

Given a particular force law: we can use the expression for required centripetal force in reverse: given a force law, what is the required velocity (hence required kinetic energy) for sustained circumnavigating motion?

In the case of Hooke's law: magnitude of the centripetal force is given by the product of a coefficient $k$ and radial distance $r$: $F=kr$

Also, to accomodate the inertial mass of a particular object the centripetal force must be proportional to inertial mass $m$.

We set the centripetal force equal to the quantity $mkr$, for the purpose of arriving at an expression for kinetic energy as a function of radial distance.

$$ \frac{mv^2}{r} = mkr \tag{4} $$

To transform the left hand side to the expression for kinetic energy: move the factor $r$ to the right hand side, and divide both sides by 2:

$$ \tfrac{1}{2}mv^2 = \tfrac{1}{2}mkr^2 \tag{5} $$

In the case of Hooke's law we have that the potential energy is proportional to the square of the radial distance. That is, the kinetic energy and the potential energy are both proportional to $r^2$


In the case of the inverse square law of gravity: the magnitude of the centripetal force is given by:

$$ F = \frac{GMm}{r^2} \tag{6} $$

We set the required force equal to the force given by (6):

$$ \frac{mv^2}{r} = \frac{GMm}{r^2} \tag{7} $$

Transform to make the left hand side the expression for kinetic energy:

$$ \tfrac{1}{2}mv^2 = \tfrac{1}{2} \frac{GMm}{r} \tag{8} $$

The examples of Hooke's law and inverse square law of gravity indicate why in (1) there is no explicit dependence on the radial distance $r$.

The process of transforming the expression for required centripetal force to the expression for the corrresponding kinetic energy raises the power of the factor $r$ by one. That is the same as what happens in the process of integrating the force to obtain the expression for the potential energy: the integration with respect to the radial distance $r$ raises the power of $r$ by 1. It follows: (1) is valid for any force law.

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The Virial Theorem holds for any system of masses where the potential energy $U$ of the interaction is a homogeneous function of the coordinates (i.e. $U(\alpha \cdot \vec{r})=\alpha^k\cdot U(\vec{r})$, with $k$ the degree of homogeneity of the function) and where the motion takes place in a finite region of space. This is shown in detail in §10 of the book 'Mechanics' by Landau and Lifshitz. A slightly more popular approach (which however also is based on the derivation in the book by Landau and Lifshitz) is given on this web page.

The Virial Theorem states then that

$$2\bar{T}=k\cdot \bar{U}$$

where $\bar{T}$ and $\bar{U}$ are the time averaged total kinetic and total potential energy repectively.

For the gravitational and Coulomb interaction, we have here $k=-1$ whereas for the harmonic oscillator $k=2$ (only in these two cases are actually closed orbits possible).

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