Return to Answer

This seems broader than the other question which was interpreted mainly in the “category theory” sense (1.).

An early, maybe earliest,^a case of sense (4.) “normal form” is Jacobi (1837) calling canonical the “Hamilton form”^b 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^c 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. Sylvester in another context (1851, p. 190) attributes the phrase to Hermite, presumably (1854):

I now proceed to the consideration of the more peculiar branch of my inquiry, which is as to the mode of reducing Algebraical Functions to their simplest and most symmetrical, or as my admirable friend M. Hermite well proposes to call them, their Canonical forms.

3. Similar: the Jordan canonical form^d and Frobenius rational canonical form^e of a matrix.

4. So far as I can tell, Jacobi may well have originated the phrase “normal form” too (1845, 1850).

^a one can find “ad formam canonicam” at least once in Euler: De reductione formularum integralium ad rectificationem ellipsis ac hyperbolae (1766, p. 28); also often aequatio canonica.
^b notoriously used before by Lagrange and Poisson.
^c called canonical by Frobenius (1877), canonical or normal by Darboux (1882).
^d called canonical by Jordan (1870, p. 114); Kronecker (1874) bitterly deplored the terminology.
^e called normal by Frobenius (1879, pp. 207-208).

This seems broader than the other question which was interpreted mainly in the “category theory” sense (1.).

An early, maybe earliest,^a case of sense (4.) “normal form” is Jacobi (1837) calling canonical the “Hamilton form”^b 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^c 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. Sylvester in another context (1851, p. 190) attributes the phrase to Hermite, presumably (1854):

   I now proceed to the consideration of the more peculiar branch of my inquiry, which is as to the mode of reducing Algebraical Functions to their simplest and most symmetrical, or as my admirable friend M. Hermite well proposes to call them, their Canonical forms.

3. Similar: the Jordan canonical form^d and Frobenius rational canonical form^e of a matrix.

4. So far as I can tell, Jacobi may well have originated the phrase “normal form” too (1845, 1850).

^a one can find “ad formam canonicam” at least once in Euler: De reductione formularum integralium ad rectificationem ellipsis ac hyperbolae (1766, p. 28); also often aequatio canonica.
^b notoriously used before by Lagrange and Poisson.
^c called canonical by Frobenius (1877), canonical or normal by Darboux (1882).
^d called canonical by Jordan (1870, p. 114); Kronecker (1874) bitterly deplored the terminology.
^e called normal by Frobenius (1879, pp. 207-208).

This seems broader than the other question 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 the “Hamilton form”^a 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^b 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^c and Frobenius rational canonical form^d of a matrix.

3. So far as I can tell, Jacobi may well have originated the phrase “normal form” too (1845, 1850).

^a notoriously used before by Lagrange and Poisson.
^b called canonical by Frobenius (1877), canonical or normal by Darboux (1882).
^c called canonical by Jordan (1870, p. 114); Kronecker (1874) bitterly deplored the terminology.
^d called normal by Frobenius (1879, pp. 207-208).

This seems broader than the other question which was interpreted mainly in the “category theory” sense (1.).

An early, maybe earliest,^a case of sense (4.) “normal form” is Jacobi (1837) calling canonical the “Hamilton form”^b 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^c 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. Sylvester in another context (1851, p. 190) attributes the phrase to Hermite, presumably (1854):

I now proceed to the consideration of the more peculiar branch of my inquiry, which is as to the mode of reducing Algebraical Functions to their simplest and most symmetrical, or as my admirable friend M. Hermite well proposes to call them, their Canonical forms.

3. Similar: the Jordan canonical form^d and Frobenius rational canonical form^e of a matrix.

4. So far as I can tell, Jacobi may well have originated the phrase “normal form” too (1845, 1850).

^a one can find “ad formam canonicam” at least once in Euler: De reductione formularum integralium ad rectificationem ellipsis ac hyperbolae (1766, p. 28); also often aequatio canonica.
^b notoriously used before by Lagrange and Poisson.
^c called canonical by Frobenius (1877), canonical or normal by Darboux (1882).
^d called canonical by Jordan (1870, p. 114); Kronecker (1874) bitterly deplored the terminology.
^e called normal by Frobenius (1879, pp. 207-208).

This seems broader than the other question 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 the “Hamilton form”^a 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 form^b of the 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 and Frobenius rational canonical form of a matrix.

3. So far as I can tell, Jacobi may well have originated the phrase “normal form” too (1845, 1850).

^a used before by Lagrange and Poisson.
^b called canonical by Frobenius (1877), canonical or normal by Darboux (1882).

This seems broader than the other question 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 the “Hamilton form”^a 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^b 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^c and Frobenius rational canonical form^d of a matrix.

3. So far as I can tell, Jacobi may well have originated the phrase “normal form” too (1845, 1850).

^a notoriously used before by Lagrange and Poisson.
^b called canonical by Frobenius (1877), canonical or normal by Darboux (1882).
^c called canonical by Jordan (1870, p. 114); Kronecker (1874) bitterly deplored the terminology.
^d called normal by Frobenius (1879, pp. 207-208).