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I often encounter the term canonical in my study of physics. What does it mean? There is canonical momentum, canonical transformations and I have even heard the phrase 'proving something more canonically'. What does the word mean in each of these contexts?

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    $\begingroup$ In physics it basically means “the important/standard one”, while in math it means something totally different, “the unique thing you can get without making any arbitrary choices”. $\endgroup$ – knzhou Aug 17 '18 at 9:35
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    $\begingroup$ @knzhou That's an answer, no? $\endgroup$ – Emilio Pisanty Aug 17 '18 at 11:59
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    $\begingroup$ @knzhou In math, doesn't it also mean "standardized"? You often do have to make some arbitrary choices to get something into canonical form, but the point is that the community has agreed which arbitrary choices to make so that there's no freedom left. There's nothing objectively natural about putting the longest rows at the top of a reduced-row echelon matrix instead of the bottom, or putting the ones on the superdiagonal instead of the subdiagonal in Jordan normal form. $\endgroup$ – tparker Aug 17 '18 at 14:05
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    $\begingroup$ @knzhou As the wiki article says, "canonical forms frequently depend on arbitrary choices." $\endgroup$ – tparker Aug 17 '18 at 14:09
  • $\begingroup$ @tparker I would call all those instances of “canonical” applied math, so really there are 3 separate meanings for 3 separate departments. $\endgroup$ – knzhou Aug 18 '18 at 0:12
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Even in physics, the term canonical requires a disambiguation for clarity. In the contexts you were citing, it means that it is a more general form. E.g. if you are dealing with momentum, then the canonical momentum refers to $p = p + q \bf{A}$, however, momentum in a Newtonian physics course would most certainly refer to $p=mv$, thus, a professor might call momentum canonical to clarify that he does not mean the more simple version, but the more general version.

For other uses of canonical, see Wikipedia's disambiguation below:

https://en.m.wikipedia.org/wiki/Canonical

Hope this clears up your answer!

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  • $\begingroup$ Isn't the generalized momenum referred to as the canonical momentum, as opposed to mv? $\endgroup$ – Abhirup Mukherjee Aug 17 '18 at 13:06
  • $\begingroup$ Yep! Sorry, I edited my answer to be more clear. $\endgroup$ – RJP Aug 17 '18 at 13:12
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Sometimes it just means "official" or "standardized" or "really important", but usually it has the more precise meaning "relating to the Hamiltonian formulation of classical mechanics". The canonical momenta are usually first introduced in the Lagrangian framework, but they are the momenta that appear in the phase space of Hamiltonian mechanics. Canonical transformations are symmetries of that phase space that preserve the symplectic structure. Canonical perturbation theory is formulated within Hamiltonian mechanics. The canonical commutation relations are a quantized version of Poisson brackets (as per Dirac's quantization rule).

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  • $\begingroup$ 'Canonical transformations are symmetries of that phase space that preserve the symplectic structure.' could you explain this a bit? $\endgroup$ – Abhirup Mukherjee Aug 17 '18 at 14:08
  • $\begingroup$ @AbhirupMukherjee The "symplectic structure" basically means the values of all the Poisson brackets between the various quantities. A change of variables is a "canonical transformation" if the expressions for the Poisson brackets of the new variables are the same as for the old variables. $\endgroup$ – tparker Aug 17 '18 at 14:52
  • $\begingroup$ So the definition of canonical transformation is one that preserves the Poisson brackets. But then, isnt there any general meaning to this word? One that encompasses all its usages? $\endgroup$ – Abhirup Mukherjee Aug 17 '18 at 18:23
  • $\begingroup$ @AbhirupMukherjee Well, as I said in my answer, they're all related to Hamiltonian mechanics... $\endgroup$ – tparker Aug 17 '18 at 19:04

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