This seems broader than the [other question](https://mathoverflow.net/questions/19644/what-is-the-definition-of-canonical) which was interpreted mainly in the “category theory” sense (1.).

An early, maybe earliest, case of the “normal form” sense (4.) is Jacobi (1837) calling [canonical](//hsm.stackexchange.com/questions/5666/why-are-canonical-coordinates-canonical) the “Hamilton form”<sup>a</sup> of the equations of mechanics, and also any variables, coordinates or “elements” in which they take this form; today we would speak of Darboux coordinates or Darboux normal form<sup>b</sup> of a symplectic structure. Remarks:

1. A big difference with the case of fractions is that the normal coordinates (or isomorphism to normal form) are far from unique.

2. Similar: the Jordan [canonical form](//en.wikipedia.org/wiki/Jordan_normal_form)<sup>c</sup> and Frobenius [rational canonical form](//en.wikipedia.org/wiki/Frobenius_normal_form)<sup>d</sup> of a matrix.

3. So far as I can tell, Jacobi may well have originated the phrase “normal form” too ([1845](//books.google.com/books?id=KpxGAAAAcAAJ&pg=PA224&dq=formam+normalem), [1850](//books.google.com/books?id=NoNKAAAAYAAJ&pg=PA479&dq=%22forme+normale+ou+canonique%22)).

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<sup>a</sup> notoriously used before by Lagrange and Poisson.<br>
<sup>b</sup> called canonical by Frobenius ([1877](//zbmath.org/?q=an:09.0249.03)), canonical or normal by Darboux ([1882](//zbmath.org/?q=an:14.0294.01)).<br>
<sup>c</sup> called canonical by Jordan ([1870](//archive.org/details/traitdessubsti00jorduoft/page/114), p. 114); Kronecker ([1874](//archive.org/details/werkehrsgaufvera01kronuoft/page/367)) bitterly deplored the terminology.<br>
<sup>d</sup> called normal by Frobenius ([1879](//archive.org/details/journalfrdierei37crelgoog/page/n216), pp. 207-208).