# What is the origin of “normal” in normal coordinates and normal modes?

I am trying to understand why vibrational modes of polyatomic molecules are called "normal" mode of vibrations and with corresponding normal coordinates. What is the origin of the term normal here? I am sure this term must have been borrowed from mechanics. What other mechanical oscillations are possible which may not be considered normal? The unabridged Oxford English Dictionary attributes early usage of word normal to W. Thomson & P. G. Tait Treat. Nat. Philos. 1867 I. ii. 274 There are in general..i distinct determinate displacements, which we shall call the normal displacements, fulfilling the condition, that if any one of them be produced alone, and the system then left to itself for an instant at rest, this displacement will diminish and increase periodically according to a simple harmonic function of the time, and consequently every particle of the system will execute a simple harmonic movement in the same period. Thanks.

• My guess would be "normal" as in "orthogonal" , in the mathematical sense. – Carl Witthoft Oct 5 '18 at 12:11
• I feel there is no orthogonal sense in vibrations. Francois suggestion seems reasonable. However, in Tisza's (one the early papers on molecular vibrations in 1930s) its calls "Infolge der Symmetrie können die Normalschwingungen entartet sein..." or Im Falle eines stabförmigen Moleküls hat die Rotation zwei Freiheitsgrade, und wir haben 3N ‒ 5 Normalkoordinaten" so it seems Germans adapted the normal coordinates as it is. The Hauptkoordinaten is less mysterious than normal coordinate or or normal modes of vibrations. – M. Farooq Oct 5 '18 at 15:37

I believe this refers to the normal form (sum of squares) a quadratic force function (a.k.a. potential) takes in such coordinates. E.g. Netto (1893, pp. 266, 271) implies as much. Stäckel (1908, p. 482) spells out the relation of general to normal (a.k.a. principal) oscillations:

By a proposition in the theory of quadratic forms ... one can find $$r$$ linear functions $$s_1,\dots,s_r$$ of $$q_1,\dots,q_r$$ such that simultaneously $$\sum A_{\alpha\beta}q_\alpha q_\beta=\sum s_\alpha^2,\qquad\sum B_{\alpha\beta}q_\alpha q_\beta=\sum \rho_\alpha^2s_\alpha^2\quad\dots$$ On introducing these principal coordinates $$s_\alpha$$ 95) the equations (1) get replaced by the simple equations: $$\ddot s_\alpha+\rho_\alpha^2\,s_\alpha=0, \tag2$$ satisfied by $$s_\alpha=\mu_\alpha\sin(\rho_\alpha t+\sigma_\alpha)$$... A general oscillation of the system is then the result of superposing $$r$$ principal oscillations (harmonic oscillations) with periods $$\frac{2\pi}{\rho_1},\dots,\frac{2\pi}{\rho_r}$$.96) ... The resulting motion is not in general periodic; for it to be, the ratios of $$\rho_1,\dots,\rho_r$$ must be integers.

95) Lagrange essentially has the principal coordinates already; they are also called harmonic, simple, normal coordinates.
96) The principle of superposition of oscillations is due to Daniel Bernoulli, Berlin Mém. année 1753, p. 173: in any system the alternating motions of the bodies are always a mixture of simple, regular and permanent oscillations of different kinds; see also D. Bernoulli, Petersburg Nov. Comment. 19 ad annum 1775, p. 239.

Attribution of the terminology to Thomson and Tait is confirmed by Lamb (1907, p. 223), Pockels (1891, p. 44) and Ball (1869, p. 597). It could have been inspired also by the “orthogonality” of the principal axes (e.g. in a different but related context, Thomson (1856, p. 487) had already written: “Def. A normal system ... is one in which the strains or stresses of the different types are ... mutually orthogonal”), but I share the impression that perpendicularity relative to general quadratic forms in higher dimensions didn’t become “intuitive” until later:

$$\hspace{4.5cm}$$ Remarks: 1) Normal form has an earlier tradition going back to Jacobi (1845, 1848, 1876), Hesse (1865), Kronecker (1866). 2) Thomson and Tait themselves don’t say normal coordinates. The first to do so may well be Lipschitz (1870) who uses Normalvariabelen (and Normalform, Normaltypus).

• I think your line of reasoning is correct. I also found this paper Some General Theorems relating to Vibrations. J. W.STRUTT, M.A (1873). "It may be said to depend on the characteristic property of the normal coordinates namely, their power of expressing the energy of the system as a sum of squares only." – M. Farooq Oct 8 '18 at 15:41
• @ Francois, I will be writing a short chemistry educational article. I will acknowledge your assistance. – M. Farooq Oct 12 '18 at 0:34

This term was borrowed from mechanics, but it is not specific to mechanics. In the original use norma was "carpenter's square", by 1500 extended generally to "rule, pattern" (c. 1500), with normal meaning "typical, common". By 1640-s a more specific use split off, "made according to a carpenter's square", and from that the modern use as "orthogonal, perpendicular" derives. These are the broad and the narrow meanings prevailing today. The broad one comes in two shades, as "generic" and as "patterned", "rule-governed". Online Etymology Dictionary continues:

"Meaning "conforming to common standards, usual" is from 1828, but probably older than the record [Barnhart]. As a noun meaning "usual state or condition," from 1890. Sense of "normal person or thing" is from 1894."

The normal distribution got its name at about the same time, based on the broad meaning, as both something standard and well regimented. In Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics we read:

"According to Kruskal & Stigler, the term normal was used, apparently independently, by Charles S. Peirce (1873) in an appendix to a report of the US Coast Survey (reprinted in Stigler (1980, vol. 2), Wilhelm Lexis Theorie der Massenerscheinungen in der menschlichen Gesellschaft (1877) and Francis Galton 'Typical laws of heredity' (1877). Of the three, Galton had most influence on the development of Statistics in Britain and, through his ‘descendants’ Karl Pearson and R. A. Fisher, on Statistics worldwide. In the 1877 article Galton used the phrase "deviated normally" only once (p. 513)--his name for the distribution was "the law of deviation." However in the 1880s he began using the term "normal" systematically.

Later Pearson seemed to imply that he had introduced the term: "Many years ago I called the Laplace-Gaussian curve the normal curve ..." (Biometrika, 13, (1920), p. 25). While Pearson did not introduce the term, it is fair to say that his "consistent and exclusive use of this term in his epoch-making publications led to its adoption throughout the statistical community." (DSB) Curiously a main theme in Pearson’s scientific work was that data did not ‘normally’ follow this distribution and that alternative distributions had to be devised."

To Tait it mattered, apparently, that "normal displacements" when left alone evolved "according to a simple harmonic function". But it could not have hurt in the eventual spread of the term that the things are commonly occurring, and in many mechanical problems "normal modes" are indeed orthogonal to each other. So all three shades blended well in this context.