Talk:Dimensional analysis

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.

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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".

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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.

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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.

No merge

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

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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)

Can we have a consistent format in the article for depicting dimensions?

If this is going to be the "truthiness":

The dimension of a physical quantity can be expressed as a product of the basic physical dimensions mass, length, time, electric charge, and absolute temperature, represented by sans-serif symbols M, L, T, Q, and Θ ...

Then let's stick to that convention throughout the entire article. I see the same dimensions expressed in italics: M, L, T, Q... and I see something that looks like dimensions expressed as: [M], [L], [T], ...

All in the same article. Can we make this article self-consistent? And then, what convention shall we use? I don't care what convention, but it should be consistent from the beginning of the article to the end. 65.183.156.110 (talk) 21:38, 10 April 2015 (UTC)