# Why are canonical coordinates canonical?

Canonical coordinates are

coordinates $q_i$ and $p_i$ in phase space that are used in the Hamiltonian formalism. The canonical coordinates satisfy the fundamental Poisson bracket relations: $$\{q_i,q_j\} = 0, \quad \{p_i,p_j\} = 0, \quad \{q_i,p_j\} = \delta_{ij},$$

-Wikipedia.

From this it is clear that we get canonical transformations and, with quantum mechanics, canonical commutation relations.

When did these coordinates start being called canonical and what was the reasoning?

Such coordinates were called canonical because they are those in which equations of motion (or, of the hamiltonian flow of a function $H$) take the “canonical form” $$\frac{dq_i}{dt}=\frac{\partial H}{\partial p_i}, \qquad \frac{dp_i}{dt}=-\frac{\partial H}{\partial q_i}$$ first written by Poisson (1809, pp. 272, 313), Lagrange (1810, p. 350), and Hamilton (1835, p. 98). Actually, none of them called that form canonical yet: according to the Encyklopädie (1935, p. 573) the first person to do so was Jacobi (1837, pp. 65-66):

On trouve au moyen de ce théorème, par le calcul même, des éléments dont les valeurs différentielles, dans le mouvement troublé, prennent la forme simple qu'elles ont dans le théorème, forme que je désigne dans mon mémoire sous le nom de canonique.

... and Thomson-Tait (1867, p. 254) commented:

This is the celebrated “canonical form” of the equations of motion of a system, though why it has been so called it would be hard to say.

As to the practice of calling the coordinates or “elements” $p_i, q_i$ themselves “canonical”, it seems to originate also with Jacobi in the aforementioned memoir, completed in 1838 but only published posthumously (1862, p. 128; German translation: 1906, p. 153):

Systema elementorum, quae in modum praecedentium per aequationes differentiales canonicas determinantur, et ipsum dicere convenit canonicum elementorum systema.

While this practice did not universally catch on at first (Thomson-Tait, Poincaré (1893), Whittaker (1917) or the Encyklopädie call the equations canonical, but not the variables), it was adopted by e.g. Donkin (1862, pp. v, 550), Tisserand (1868, p. 258; 1889, p. 164), Schering (1873, p. 23), Lie (1874, p. 258), Routh (1892, pp. 304-306), Dziobek (1892, pp. 102-103), Charlier (1902, pp. 56-58), Dirac (1925, p. 651), etc. In fact it must have leaked somehow between 1838 and 1862, for Cayley has it already in (1858, p. 9):

18. There is, however, one important point which requires to be adverted to. Lagrange, in the memoir of 1810, and the second edition of the ‘Mécanique Analytique,’ remarks, that for a particular system of arbitrary constants, viz., if $\alpha,...$ denote the initial values of the coordinates $\xi,..$ and $\lambda,..$ denote the initial values of $\smash[b]{\frac{dT}{d\xi'}},...$ then the equations for the variations of the elements take the very simple form $$\frac{d\alpha}{dt}=-\frac{d\Omega}{d\lambda}..,\quad \frac{d\lambda}{dt}=\frac{d\Omega}{d\alpha},...$$ This is, in fact, the original idea and simplest example of a system of canonical elements; viz. of a system composed of pairs of elements, $\alpha, \lambda$, the variations of which are given in the form just mentioned.

• +1 for the huge work of finding and linking all the references. – Antoine Feb 13 '17 at 18:05

I was told they are called "canonical" because of their prevalence and comparative simplicity. I believe the etymology is from the Greek "kanon", meaning "standard" or "model", or "usual".

I was told by the professor in a graduate physics course that the equations were called "canonical" because they were so perfect that they could be laws of the church, that is canon laws. At the time of their creation, the church was the most powerful entity extant. Thus, they were perfect enough to be church laws or "canonical" laws.

• That just sounds like an explanation of where the word "canonical," which is not a term in physics only, comes from. It does not sound like an authoritative explanation of its use in this situation rather than its general etymology. – KCd Jan 26 '18 at 15:59

A more appropriate translation of canon (kanon) is "law". Consider for example the dictionary definition of "canonical hours": Canonical hours: certain stated times of the day, fixed by ecclesiastical laws, and appropriated to the offices of prayer and devotion;

• Currently, this post does not answer the question as stated. Could you perhaps expand it to do so? – Danu May 20 '17 at 21:05