# Talk:Dimensional analysis

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## Compound units / ampere

"For example, an ampere is a measure of electrical current, which is fundamentally electrical charge per unit time and is measured in coulombs (a unit of electrical charge) per second, so 1A = 1C/s."

Could this be expressed in another way? 1A is the defining quantity in the SI system, so by definition 1C = 1As. This gives a circular argument: 1A = 1C/s = 1As/s = 1A. I agree that the physics are correctly described, but it's not how the SI system works. Jens 130.243.105.40 (talk) 09:44, 8 August 2014 (UTC)

## Old talk

In the most primitive form, dimensional analysis is used to check the correctness of algebraic derivations: in every physically meaningful equation, the dimensions of the two sides must be identical. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers, which is typically achieved by multiplying a certain physical quantity by a suitable constant of the inverse dimension.
I do not agree. Dimensional analysis is used to solve PDEs. The statement just describes e.g., stochiometry.

I admit that I don't know how to use dimensional analysis to solve PDE's (do you have any references?), but this paragraph was really just the beginning, showing the most primitive "dimensional analysis" as taught in college chemistry classes: make sure that the dimensions are right. I agree there's much more to Dimensional Analysis than that, and the rest of the article shows it, so I think the criticism is not justified. --AxelBoldt

What I was trying to say is the "monorail" algorithm for using units to solve stochiometry problems is not really dimensional analysis, but to be fair I will start cracking some books on this.

The above mentioned reduction of variables uses the Buckingham Pi theorem as its central tool. This theorem describes how an equation involving several variables can be equivalently rewritten as an equation of fewer dimensionless parameters, and it provides a method for computing these dimensionless parameters from the given variables, even if the form of the equation is still unknown. Two systems for which these parameters coincide are then equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one.
This is not quite correct either. The resulting dimensionless parameters generally need to be determined experimentally, or there must be some sort of experimentally verified constitutive relationship. No one as yet can predict a Froude or Mach number, we can only measure them.

That's what I was trying to say: the Pi theorem tells you how to turn the measured variables into dimensionless parameters, and then you have to empirically find the relationship between those dimensionless parameters. No one can predict a Mach number, but people can predict the proper formula for Mach numbers. How can we clarify the above paragraph? --AxelBoldt

I think you mean "proper units" for Mach numbers...
That's the 64 dollar question. People that know how to do this (e.g., Barenblatt) just smile enigmatically when asked "how you do dat?" The best that I have been able to determine is that the process is like that cartoon of the physicist at the blackboard, where in a long chain of formulas, the one in middle is labeled "magic here".

I removed the "typed family of fields" comment, since there is no such thing in mathematics.

Such a thing can be well-defined. See G. W. Hart "Multidimensional Analysis".

Also,

Note also that the dimensionless numbers are not really dimensionless. The actual

structure of a dimensionless number is unity in the type. For example, consider the so-called dimensionless unit of strain: L/L. The L/L units are usually dropped, either implicitly or explicitly, but it is a mistake to regard strain as a physically meaningful quantity without some notion of the L in the denominator, which acts as a gauge length. For another example, consider the physical meaning (none)of adding strain (dimensionless) to Mach (dimensionless).

I don't understand this. Are you arguing that even dimensionless numbers should keep their dimensions? I can't make mathematical sense of that. Is L/L a different unit in your system than M/M? --AxelBoldt

It's not "my system", it's physics. L/L is a different unit than M/M. Yes, I am saying that dimensionless numbers should keep their dimensions. Think about it carefully. The real numbers used for computing physical quantities are meaningless without units. Velocities must be expressed in terms of L/T, whether it be meters/sec or furlongs/fortnight. The problem is that while the real numbers obey the axioms for a field, units obey group axioms. We can do math on the reals alone (analysis), the units alone (group theory) or real numbers with units attached to each quantity (typed family of fields). In the scalar world all of this is pedantic frippery. However, linear systems constructed to solved differential equations describing matter will have units attached. As it turns out, with care, units may be mixed within the system, and a solution determined using LU decomposition (say) will remain dimensionally correct. You can integrate this stuff too... (heh heh) If strain didn't keep its L/L dimensions, then strain energy per unit volume (FL/L^3) would just be F/L^2 which is units of stress.

No further comments or analysis tolerated on dimensional analysis

Can't you see how many people wrote about it?

• Barenblatt, G. I., "Scaling, Self-Similarity, and Intermediate Asymptotics", Cambridge University Press, 1996
• Bridgman, P. W., "Dimensional Analysis", Yale University Press, 1937
• Langhaar, H. L., "Dimensional Analysis and Theory of Models", Wiley, 1951
• Murphy, N. F., Dimensional Analysis, Bull. V.P.I., 1949, 42(6)
• Porter, "The Method of Dimensions", Methuen, 1933
• Boucher and Alves, Dimensionless Numbers, Chem. Eng. Progress, 1960, 55, pp.55-64
• Buckingham, E., On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis, Phys. Rev, 1914, 4, p.345
• Klinkenberg A. Chem. Eng. Science, 1955, 4, pp. 130-140, 167-177
• Rayleigh, Lord, The Principle of Similitude, Nature 1915, 95, pp. 66-68
• Silberberg, I. H. and McKetta J. J., Jr., Learning How to Use Dimensional Analysis, Petrol. Refiner, 1953, 32(4), p179; (5), p.147; (6), p.101; (7), p. 129
• Van Driest, E. R., On Dimensional Analysis and the Presentation of Data in Fluid Flow Problems, J. App. Mech, 1946, 68, A-34, March
• Perry, J. H. et al., "Standard System of Nomenclature for Chemical Engineering Unit Operations", Trans. Am. Inst. Chem. Engrs., 1944, 40, 251
• Moody, L. F., "Friction Factors for Pipe Flow", Trans. Am. Soc. Mech. Engrs., 1944, 66, 671

Who the heck do you think you are?

little guru

## Restriction on functions

This whole section (except the matrix part) is treated in two other places in the article, and is a bit of overkill. The bit about certain matrices, while true, is very much of a sideshow curiosity, and is not usually used in dimensional analysis. The section also destroys the flow of the article, and seems to be at the very least, out of place. I am deleting it for now, but would be willing to reconsider if there is a strong objection.

## Commensurability

"Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although torque and energy share the dimension ML2/T2, they are fundamentally different physical quantities."

Is there a way to reconcile this fact with commensurability? Can the consistency of dimensional analysis be maintained by say changing the dimension of torque (to perhaps include an orientation, or a direction) or something else, or is this a fundamental problem somehow? Robleroble (talk) 22:31, 13 March 2013 (UTC)

Yes, you can do that by assigning a dimension to angles. Torque times a rotation angle is energy, but because angles are conventionally dimensionless, a torque has the same dimension as energy. But this same problem will then crop up elsewhere, so it is still a fundamental problem. The issue is that physics is fundamentally dimensionless, the units we use are ultimately arbitrary human constructs.
The reason why dimensional analysis works is because it is a scaling argument in disguise. Given a number of physical quantities, you can always make them dimensionless by multiplying them with combinations of c, hbar and G, then you can apply any mathematical function to those variables, and then you can give that any dimensions you like by multiplying this by an appropriate combination of c, G and hbar. So, a priori, any physical quantity Y could be related to some given physical quantitities x1, x2, x3, ... via any given function, if you are allowed to use the constants c, G and hbar. So, dimensional analysis only yields non-trivial results if you make the additional demand that you are not allowed to use one or more of the constants hbar, c, and G. But this means that you are looking at e.g. the classical limit hbar to zero, or the nonrelativistic limit c to infinity, which can be interpreted as looking at the physics at certain scaling limits.
It is then only in these scaling limits that certain variables are incompatible. E.g. you can compare lenghts to time intervals, the conversion factor is c. But you lose this conversion factor precisely in the scaling limit that corresponds to the classical limit. Count Iblis (talk) 23:55, 13 March 2013 (UTC)
Check out the "extensions" section. In Siano's extension, the energy is without direction ($\mathbf{1}_0$) while, for example, the x-component of torque is directed, having dimension proportional to $\mathbf{1}_x$. This is essentially what Count Iblis is saying. In Siano's extension, angles have a direction. For example, the direction of an angle formed by 2 lines in the xy plane has a dimension proportional to $\mathbf{1}_z$ PAR (talk) 00:05, 14 March 2013 (UTC)
I've often wondered why we don't simply treat angles dimensionally. I've even come across a couple peer reviewed academic papers that are in favor of this [1] [2]. The most obnoxious part about treating angles dimensionally seems to be that we would have to divide by 1 radian constantly. For example, sin(θ) becomes sin(θ/rad) and exp(iθ) becomes exp(iθ/rad). I suppose we could just define a physical constant c=1 rad=180/π deg=1/(2π) turn, but then this would pop up everywhere. Are there some other unintended effects that would prevent me from simply treating angles like a dimension?
[1] K. R. Brownstein. Angles---Let's treat them squarely. American Journal of Physics. July 1997. Volume 65, Issue 7, pp. 605. [DOI].
[2] Jean-Marc Lévy-Leblond. Dimensional angles and universal constants. American Journal of Physics. September 1998. Volume 66, Issue 9, pp. 814. [DOI].
--Mk29 (talk) 14:20, 16 August 2013 (UTC)

## No merge

"Dimensional analysis" is a discipline, "units of measurement" is a standard. NO MERGE VOTE.

The old alert was correct, a "need to sync": Wikipedia:Summary style

{{sync|Units_of_measurement#Calculations_with_units_of_measurements}}

## Did Newton refer to dimensional analysis as the "great principle of similitude"?

The article asserts that Newton (in 1686) would have called the method of dimensional analysis "the great principle of similitude". Reference is given as "Walter R. Stahl, Dimensional analysis in mathematical biology, Bulletin of mathematical biophysics, Vol. 23, 1961, p. 355". - The reference to Stahl is correct, the reference to Newton is not. Nowhere did Newton ever say anything like that, or similar to that assertion. The assertion is simply false. Never cite secondary sources! 91.37.165.191 (talk) 19:51, 8 September 2014 (UTC)

As a wikipedia contributor, I'm probably not qualified (and haven't time) to determine from primary sources (couched in 17th century conceptual-framework and terminology) whether your assertion is true (regarding everything Newton ever wrote). I also can't verify your qualifications (and compare them to mine or Stahl's). What I can do is verify secondary sources. So now we do know for a fact that a scientific journal (math. biophys.) published an expert's assertion that Newton (in Principia Mathematica, 2 §7 (1686)) indeed referred to dimensional analysis as the Great Principle of Similitude. Other sources (including [1] and [2]) give further discussion of Newton's and other's historical contributions. Cesiumfrog (talk) 09:09, 9 September 2014 (UTC)

Thank you. In this case, instead of attributing Stahl's assertion simply to Newton as a fact, wikipedia should perhaps tell the reader the truth that "Stahl asserted that Newton would have called ..." etc. etc. 91.37.155.29 (talk) 12:28, 9 September 2014 (UTC).

I add that I have checked the "sources" you have given me, to no effect. To say it once again: The assertion that Newton in Principia, Book II, Prop. 32 (probably the "§ 7" you refer to, but Newton has no "paragraphs" but "sections") would deal with "dimensional analysis" explicitly or implicitly, speaking of a "great principle of similitude", is simply not true. It is also not true what your source "Brennan" asserts on p. 44 of his book, when he refers to Newton's Principia, Book II, prop.32. No trace of "distinct entities as length, inertia, and mass" to be found there. Nowhere speaks Newton of "concepts" as Brennan asserts. I propose to correct the page by cancelling the mistaken reference to Newton.91.37.155.29 (talk) 14:52, 9 September 2014 (UTC)

Since your interpretation of principia appears to disagree with those of all published experts, why not first submit your argument and explanation to a peer-reviewed history-of-science journal? Or in cases such as this (whenever one wikipedian disputes some mainstream consensus) do you think it would be a better practice if we immediately rewrote the article to say "Stahl, Brennan, West, and others, are all wrong according to an anonymous editor and the concepts underlying dimensional analysis were completely unimagined and unprecedented before the 19th century"? Cesiumfrog (talk) 12:11, 10 September 2014 (UTC)

Please will you note that I'm not "interpreting the principia". Also, I do not assert what you impute to me, that "the concepts underlying dimensional analysis were completely unimagined and unprecedented before the 19th century". Rather, I'm telling you that something which you quote (!) from Principia, Book II, Sect. 7, cannot be found there, no matter how many "published experts" assert the same nonsense. You can most easily verify that I'm right. So I'm pointing to an evident error in the Wikipedia which should be corrected immediately, and you should perhaps be grateful to me for wasting my time in correcting your mistake. Or am I to learn that Wikipedia prefers the false opinion of "experts" over the simple truth? 91.37.163.73 (talk) 05:26, 11 September 2014 (UTC)

91, I checked this out at this reference. I'm pretty sure you're correct. And I removed the sentence. 71.169.182.51 (talk) 12:55, 11 September 2014 (UTC)
I've also checked that link, and although that translation from Latin does not use our modern terminology, it is broadly consistent with the expert summaries (for example, a principle of similarity from geometry is applied to a problem in mechanics and used to conclude on the proportionality of particular physical measurements). More discussion can be found in chapter 4 of J.C. Gibbings' Dimensional Analysis book, with further detail in the references listed at the end of that chapter (some of which include quotes from Principia). Another quote from Newton appears in the history chapter of Dimensional Analysis by Jonathan Worstell. Shouldn't our history section (like that of all these sources) begin prior to Maxwell? Cesiumfrog (talk) 08:09, 18 September 2014 (UTC)
The let's get a reference that is verifiable, unambiguous, in English, and with widely-accepted repute. I don't really have a dog in this fight, but it's just that when I checked this out from what i could find on the internet, 91's case seemed to be supported. There is no evidence, so far, that Newton said or wrote such a thing. It would be fine by me to credit Newton with the basic concept of dimensional analysis, but let's get a solid reference that no one can argue with. 64.17.96.139 (talk) 21:36, 18 September 2014 (UTC)
It's a little bit tendentious for you to ask for 'a reference' after a dozen sources (including multiple textbooks, as well as scholarly articles devoted to the history of dimensional analysis) have already been listed above. If this doesn't count as evidence for you, what does? An anonymous forum post? Cesiumfrog (talk) 04:05, 20 September 2014 (UTC)