Buckingham's pi-theorem

A note used in the course TMA4195 Mathematical modelling

I wrote the note in a fit of frustration over the apparent lack of precise proofs or references to a proof in the literature. Recently (2001-09), however, I have been made aware of the book Symmetries and Differential Equations by George W. Bluman and Sukeyuki Kumei (Springer-Verlag 1989, ISBN 0387969969), which has an extensive bibliography and discussion pertaining to this important theorem. I won't repeat their entire bibliography here, but here are a handful of references:

• Buckingham, E. On physically similar systems; illustrations of the use of dimensional equations. Phys. Rev. 4, 345-376 (1914).
• Buckingham, E. The principle of similitude. Nature 96, 396-397 (1915).
• Buckingham, E. Model experiments and the forms of empirical equations. Trans. A.S.M.E. 37, 263-296 (1915).
• Görtler, H. Zur Geschichte des pi-Theorems. ZAMM 55, 3-8 (1975). (On the history of the pi theorem, in German.)
• Curtis, W.D., Logan, J.D., Parker, W.A. Dimensional analysis and the pi theorem. Lin. Alg. Appl. 47, 117-126 (1982).

Added on 2006-09-26: Here is an other reference that, I have been assured, is good from an engineering point of view: Dimensional Analysis and Theory of Models by Henry L. Langhaar (John Wiley & Sons, Inc., 1951, but also apparently Krieger Publishing Co., 1980, ISBN 0882756826). I have not yet seen it myself.

My own note

A note on the 2004 version. The contents of this note is unchanged since 2001. What is new is the use of type 1 PostScript fonts and, in particular, the Fourier font package. Among other things, this makes the PDF versions easier to read on the screen.

Due to the above references, my own little note is perhaps not so very relevant anymore. However, it contains a couple of examples, and it is possible that it offers a somewhat different perspective than other material, since it was developed relatively independently. Or maybe not - I'll know for sure when I have the time to take a closer look at the existing literature. Without further ado, here it is:

• As a PDF file (8 pages, A5 paper size).
• As a PDF file ("two up", 4 pages, A4 paper size; landscape format)

  If you must print on US letter paper, be sure to print the PDF version and instruct your PDF printing program to shrink the page to fit the paper.

A micro-biography of Edgar Buckingham

Edgar Buckingham (1867–1940) was educated at Harvard and Leipzig, and worked at the (US) National Bureau of Standards (now the National Institute of Standards and Technology, or NIST) 1905--1937. His fields of expertise included soil physics, gas properties, acoustics, fluid mechanics, and blackbody radiation. (Unfortunately, the NIST web pages don't seem to have any information on Edgar Buckingham.)

Copyright

Please respect my copyright. I allow free copying for personal and classroom use, but otherwise no redistribution or publication in any form without specific permission.

Wikiverse link

There is a short article on Buckingham's pi-theorem at Wikiverse. It links to this page.