# Talk:Virial theorem

WikiProject Physics (Rated C-class, Mid-importance)
This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of Physics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
C  This article has been rated as C-Class on the project's quality scale.
Mid  This article has been rated as Mid-importance on the project's importance scale.

## §n a natural number?

For the power-law relationship given is the number 'n' appearing in the exponent required to be a natural number? I suppose it is, otherwise any force should be described as satisfying a power-law. But then I've read numerous papers on 1/f noise discussing a 3/2 power law relation. Mandelbrot described fractional power-laws in a paper with John van Ness in the 1950's. So what is it that makes natural number power laws so different from real number power laws?

Reply: Why is it true that any force can satisfy a power-law with a non-natural exponent? Does this just happen to be the case for every force in existance? That would surprise me, though I cannot currently think of a counter example. Certainly arbitrary functions cannot be described as power laws.
Never heard of a Taylor expansion, then?
I think $\overline{K} = \frac{n+1}{2} \overline{V}$ holds also for fractional n. It assumes $V= a r^{n+1}$, not a sum of powers.--Patrick 07:55, 13 October 2005 (UTC)
A Taylor expansion is a sum of power-laws, not a single power law. The strong force and weak force are not power laws, but drop precipitously at larger distances. The force between two magnetic dipoles is proportional to the inverse-cube of the distance, but also proportional to a factor determined by the relative orientations of the dipoles and the separation between them. 128.165.112.42 15:04, 25 May 2006 (UTC)

After the copyright situation is resolved, a possible good link to add is John Baez's explanation 128.165.112.42 15:04, 25 May 2006 (UTC)

See Landau's mechanics p. 23. If the potential is a homogeneous function of order n of position, $V(\alpha r)=\alpha^n V(r)$, then, by Euler's theorem on homogeneous functions, $-F\cdot r = \nabla U \cdot r= n U$.
Therefore, the only requirement is homogeneity, not integer n.

## §Have you heard of Copyright?

I wonder who wrote this page, but I'd like to remind him that copyright should be respected.

If you compare this page with Goldstein's "Classical Mechanics" p.70, you'd understand by point.

Besides, if you intend to copy anyway, please state "if the force is central" just before "If V is a power-law function of r". This can't be omitted, because you are using the same r to represent the position vector and the radius vector ambiguously.

I put a notice at Wikipedia:Copyright problems but I don't know the proper procedure for this. 80.203.45.67 14:00, 8 April 2006 (UTC)
Which edition? Zarniwoot 00:20, 14 May 2006 (UTC)

## §Nomenclature

I don't like the nomenclature K for the kinetic energy. Remember that K is still often used for force, too, especially in Special Relativity, given that until the 40s, German was especially important in physics. In German, force is named Kraft and the relativistic 4-force is often written as K.

For the kinetic energy, the nomenclature KE (when not E(kin) or something like that) is much better. However, E would be very good to appear, since the official term for energy is E or W (of work). I think, an often used abbreviation for kinetic energy is T. N.M.B.R.Nbez 11:06, 7 May 2006 (UTC)

## §Copyvio

I've reverted to the revision (cur) (last) 03:08, 6 May 2004 158.193.210.45, as per concerns raised above that this is a copyright violation of "Classical Mechanics". As I don't have a copy of the book, I'm unable to check what exactly is a copyright violation.

Please see this for a list of differences between that version and the pre-copyright violation noted version. I've also restored the interwiki links as well as the external links to the version right before it was noted as a copyright violation. Jude (talk,contribs,email) 03:10, 15 May 2006 (UTC)

## §No copyvio; have fun!

Hi all, I re-wrote this article from scratch without looking at any outside sources (e.g., books, articles or websites), so there should be no copyright problem now. I did keep the G notation for the virial, but that's about the only hold-over. I'm clueless about how the virial theorem has been used in other settings (e.g., for gases or in quantum mechanics), though, so I'm looking forward to your additions! :D WillowW 10:51, 9 July 2006 (UTC)

## §Intro needed?

Something about its significance? —Preceding unsigned comment added by HairyDan (talkcontribs) 22:08, 17 October 2006 (UTC)

I made an try at this; does it seem good to you? Willow 23:06, 17 October 2006 (UTC)

## §A little mistake in the etymology

This not exactly a technical issue, but for precision's sake... Although "virial" actually comes from "vis" (force, strength), the Latin substantive is highly irregular, and the genitive form is "roboris" (which was a genitive borrowed, quite curiously, from "robur, roboris", oak). The quoted genitive "viris" doesn't actually exist, and "roboris" would only be confusing... I have simply deleted the genitive.

Massimo 10:43, 18 December 2006 (UTC)

Wow, that's a surprise — thanks for catching that! I knew that vis was irregular in some cases (e.g., vim and vi) but I had so often read the nominative plural vires that I just assumed that there was a corresponding genitive singular. :( Thanks very much, Massimo; I love the way Wikipedia helps us to help each other and learn new things from each other. :) Warm wishes for the holidays, Willow (a distant relative of Quercus robur ;) 15:31, 18 December 2006 (UTC)

## §Sign conventions?

When I first learned the viral theorem, it was in the form 2T+V=0; this is because the usual convention is for the potential energy V of a bound system to be negative. It appears that the statement of the theorem in the article regards V to be of the opposite sign to the usual convention. This needs to be discussed. Bill Jefferys 03:56, 6 March 2007 (UTC)

Hi Bill,
Thanks for being so careful in cross-checking the article! The form you learned pertains to gravitational or electromagnetic potential energy and is entirely correct. But the formula in the text is also correct, since n=-1 for gravity and Coulomb's law. If you check the derivation carefully — which I would appreciate very much — I hope you'll find that no sign error has been made. Thanks again! :) Willow 10:36, 29 March 2007 (UTC)

In the section on the virial radius formula, "H" and "G" should be defined immediately before or after the formula. I'd do it myself, but I'm not sure what H is! A reference would also be good...perhaps one of the "references" or "additional readings" already listed would work? Finally, it could use a sentence giving some context as to why the virial radius is defined the way it is, and why it's a useful thing to think about (for example, is the factor of 200 arbitrary?) Steve 15:05, 16 July 2007 (UTC)

I tried to fix it some, e.g. indicating that the factor 200 is arbitrary, but I find this section rather suspect nonetheless. --Art Carlson 20:45, 18 July 2007 (UTC)

## §Who proved the virial theorem?

This is a great article, but how about a little more history? It says Clausius gave the virial theorem it's definition. Does this mean he originated it? Proved it? If not, who did? --Chetvorno 07:48, 21 August 2007 (UTC)

Yes, Claudius originated it, i.e. he derived it and presented in the form used in this article.--Zinger (talk) 11:53, 3 May 2010 (UTC)

## §KE / PE relationship

Would it be of interest for this article to reference and state simply why KE is equal to PE for a pendulum, spring, water wave, etc., and why KE is not equal to PE for orbital mechanics, Bohr atom, etc.? 50MWdoug (talk) 07:24, 31 March 2008 (UTC)

I suspect that the reason is the virial theorem is about kinetic energy and potential energy between the particles, e.g. gravitational forces; these forces are dependent on the square of the distance between the particles whereas the pendulum bob is assumed (constrained) to be in a uniform gravitational field, so the gravitational force on the bob is taken to be constant with height.
If you are still watching this page I would be intersted in your response. --Damorbel (talk) 08:06, 6 April 2013 (UTC)

## §$n \langle V_{TOT} \rangle = \langle V_{TOT} \rangle$?

I believe that the statement $2 \langle T \rangle = n \langle V_{TOT} \rangle$ in the beginning, where $V_{TOT}$ is said to be total potential energy of the system, to be correct the subscript of $V_{TOT}$ should be $V_{n}$ so that $n \langle V_{n} \rangle = \langle V_{TOT} \rangle$. Thus making $\langle V_{n} \rangle$ the time average potential energy of the particle $n$, and the sum of all such particles in the system equal to $\langle V_{TOT} \rangle$, the total time averaged potential energy of the system.

If someone could verify this to be true as well, it would be greatly appreciated.

Basilf1 (talk) 18:50, 13 May 2008 (UTC)

Here $n$ is not a number of particles but the power of distance in the expression for potential: $V(r_{jk}) = \alpha r_{jk}^n,$. And $V_{TOT}$ represents the net potential energy of the system. So the equation $2 \langle T \rangle = n \langle V_{TOT} \rangle$ is correct. --Zinger (talk) 17:21, 3 May 2010 (UTC)

## §Definition of the virial

The article currently reads, "The scalar virial G is defined by the equation $G = \sum_{k=1}^N \mathbf{p}_k \cdot \mathbf{r}_k$ ...". Subsequent parts of the derivations make references like "time derivative of the virial" which, from context, are consistent with calling this G the virial. However, I'm more familiar with the virial being defined as something like $-\frac{1}{2}\sum_{k=1}^N \left\langle \mathbf{F}_k \cdot \mathbf{r}_k \right\rangle$, or perhaps without the $\frac{1}{2}$, and maybe sometimes without the minus sign. Goldstein and and Thornton/Marion agree with thid $F\cdot r$ version. (I think T/M is just quoting Goldstein.) Landau/Lifshitz also agrees, but doesn't have the 1/2 in front. I'm not aware of any texts that define the virial in the $p\cdot r$ way we're using here. Can anyone find a reference? I haven't seen Clausis's original work myself.

I think the L/L and Goldstein references alone are sufficient to justify adding a sentence to the lead as,

"... technical definition by Clausius in 1870. Modern usage varies by author as to which quantity in the virial theorem is known as "the virial" . Common choices include $-\frac{1}{2}\sum_{k=1}^N \left\langle \mathbf{F}_k \cdot \mathbf{r}_k \right\rangle$ and 2 times this quantity." (with references to Goldstein and L/L as commonly accepted standards. If someone can quote a source with the $p\cdot r$ version, that should be mentioned here, too.)

But, then the derivations would have to be cleaned up to be consistent with the $F\cdot r$ virial instead of the current G. So, I thought I'd check for consensus before making that more extensive edit. (New editor here, please don't bite.) Spatrick99 (talk) 21:01, 2 March 2010 (UTC)

I've checked the sources on the definition of virial and your statement seems to be correct: virial should be defined like $-\frac{1}{2}\sum_{k=1}^N \left\langle \mathbf{F}_k \cdot \mathbf{r}_k \right\rangle$. The only mentioning of the virial in the form $G = \sum_{k=1}^N \mathbf{p}_k \cdot \mathbf{r}_k$ could be found on the page which link is located above in the discussion: John Baez's explanation of virial. But the author uses Goldstein as a reference, where virial is explicitly defined as the former. So, I agree on changing the definition.

But that should not lead to any changes in the rest of the text (except where G is mentioned as a virial). Virial theorem could be proven even without introducing "virial" at all. --Zinger (talk) 12:52, 3 May 2010 (UTC)

## §Why $\left\langle \frac{dG}{dt} \right\rangle_\tau = 0$

I think the article isn't clear enough on it.88.159.72.240 (talk) 01:00, 8 March 2010 (UTC)

## §Why is moment of inertia introduced?

The article defines a scalar moment of inertia before defining the virial, but doesn't establish any meaningful connection between the two. Should the use of the moment of inertia be removed? If not, can it be better justified as a related idea? 70.247.169.94 (talk) 02:19, 4 July 2010 (UTC)

Sure it does. The third equation in Section "Definitions of the virial and its time derivative" relates the two quantities, and the section "Inclusion of electromagnetic fields" discusses the relationship in the text as well. --Art Carlson (talk) 19:32, 4 July 2010 (UTC)

## §Propose to remove mention of $I$ (moment of inertia [sic])

A moment of inertia is defined relative to an axis of rotation, rather than relative to the origin. For instance, if the rotation is about the z-axis then the contribution of a mass $m$ at position $(x,y,z)$ is $mr^2$ where $r^2 = x^2+y^2$. In particular, $r$ is not the distance to the origin. However, the quantity $I$ in the article uses $r$ as the distance to the origin. How is $I$ a moment of inertia? I think we should remove this misleading use of $I$. —Quantling (talk | contribs) 15:19, 13 July 2011 (UTC)

Are you proposing to remove the concept and the equation, or just give it another name, or just mention that the definition may not be what one would expect? Collins on p. 9 of this reference calls it "the moment of inertia (by definition) about a point". Why shouldn't we do the same? --Art Carlson (talk) 07:44, 14 July 2011 (UTC)
I am proposing that all mention of $I$ be removed. Those equations that serve only to relate $I$ to $G$ would be removed as superfluous, and other equations that use $I$ would instead use $G$ directly. I don't see that the existence of $I$, regardless of what you name it, makes the explanation of the Virial theorem any clearer. As for Collins, he/she doesn't actually use $I$ as "a moment of inertia about a point", or even explain what the heck that could possibly mean. Is there a reference out there that uses this $I$ in such a way that could reasonably be deemed a moment of inertia? —Quantling (talk | contribs) 14:17, 14 July 2011 (UTC)
$G = \frac{1}{2} \frac{dI}{dt}$ is an elegant and useful result. A good example is given farther down, in the section on "Inclusion of electromagnetic fields", which concludes that "the lifetime of a plasmoid is expected to be on the order of the acoustic (or Alfvén) transit time." It is important to keep this.
I obviously is something like the moment of inertia, even if it is not the moment of inertia. I believe Goldstein calls it this, too, but I would have to check. When I google "moment of inertia" "virial theorem", I find a number of references that use this quantity and call it the same thing.
--Art Carlson (talk) 08:05, 15 July 2011 (UTC)
I'm still not getting it; how does the use of $I$ instead of $G$ make it clearer that "the lifetime of a plasmoid is expected to be on the order of the acoustic (or Alfvén) transit time." Also, do any of the references that use "moment of inertia about a point" get used in actual classrooms or laboratories, or is this something that got said and now bounces around among the crackpots? —Quantling (talk | contribs) 17:39, 16 July 2011 (UTC)
I don't see how you would phrase the plasmoid argument without using I.
I just checked George Schmidt, Physics of High Temperature Plasmas. He simply calls I "the moment of inertia" without any qualification. This is one of the textbooks I learned plasma physics from in graduate school. The usage is definitely mainstream.
--Art Carlson (talk) 08:44, 17 July 2011 (UTC)

## §Relevance to Stellar Evolution

From the Virial theorem, an average temperature for a star can be derived. From this, the stellar evolution can be predicted, from the expansion of a main sequence star as hydrogen is depleted up until after helium burning stops.

I think it'd be a good idea to include something on this, and I'd write it myself except I don't know enough about it to be what you'd consider an accurate source. I can't find much literature on this on the internet, but it was something we did in my Astrophysics course, and it'd be a good application of the theorem to be included in the article. — Preceding unsigned comment added by 31.205.74.33 (talk) 21:47, 24 May 2012 (UTC)

I agree. I'll try to add a section on 'Virial Theorem in Astrophysics' which will be a bit more general --- but still easily applied to stellar evolution. All Clues Key (talk) 15:40, 12 October 2012 (UTC)