Talk:Canonical transformation

Generating function G

(Traumantischer): I really like the article. It is not too hard to follow, but I got stuck at the important point where the generating function G is introduced from the argument of two disappearing variations. First the little lambda makes me wonder: It's introduced without being obvious, then it's set to 1, and it disappears in the same line where it was introduced. Then it would be helpful to know why these variations are written down (action integral?). The function G(q,p,t) then appears as generating function, but why G(q,p,t) is necessary isn't crisp and clear (even though I have my guesses about it): All I'm saying is that G(q,p,t) isn't well motivated. I think, mathematically it's all correct, but somehow this is a point where the text doesn't flow and becomes a bit mysterious. Could the author add a few words for clarification? — Preceding unsigned comment added by Traumantischer (talk • contribs) 03:31, 20 July 2007 (UTC)

guest_user: I linked at that point a wiki book, which explains a bit why lamda and G(q,p,t) are needed here and added some explaining text. But I still agree with Traumantischer it should be motivated a bit better here and maybe someone like start up with my text and integrate it a bit better in to the text flow of the article?:) — Preceding unsigned comment added by 84.56.88.3 (talk) 17:51, 24 December 2011 (UTC)

Direct approach

The "Direct approach" section is not very clear to me. It reads

[cite][...]the time derivative of a new generalized coordinate Q_m is

${\dot {Q}}_{m}=[...]\lbrace Q_{m},H\rbrace$

[...]We also have the identity for the conjugate momentum P_m

${\frac {\partial H}{\partial P_{m}}}={\frac {\partial H}{\partial \mathbf {q} }}\cdot {\frac {\partial \mathbf {q} }{\partial P_{m}}}+{\frac {\partial H}{\partial \mathbf {p} }}\cdot {\frac {\partial \mathbf {p} }{\partial P_{m}}}$

If the transformation is canonical, these two must be equal, resulting in the equations [...][/cite]

"The two must be equal" refers to the identity

$\lbrace Q_{m},H\rbrace ={\frac {\partial H}{\partial P_{m}}},$

which is true, but not trivially. I believe that some words must be spent explaining this to improve clarity. Thank you --Giuseppe Negro (talk) 19:34, 30 December 2011 (UTC)

The equation follows from invariance of Poisson bracket so that was previously circularly defined. Currently this issue is fairly well mitigated but some of the assumptions presented seem to come from nowhere. Maybe the article can be improved further. EditingPencil (talk) 22:22, 1 December 2023 (UTC)

Actually, I do not see why it should be true (but I am confused and have no idea about the Hamilton formalism). Recall that we want P and Q to be canonical with respect to some new K (not to the old H). So why should they satisfy the equations for H!? As an example (copied from Landau Lifschitz): Set Q=q, P=2p. Then it seems that K(P,Q)=2H(p,q)=2H(P/2,Q) witnesses that $Q,P$ is canonical as defined in this wiki article (but it will not satisfy Liouville's Thm, so it will not satisfy the given set of equations). It seems that Landau Lifschitz only calls a transformation canonical if it comes from a generating function (which is a stronger property than the "there is a K such that..." as defined in the wiki article, as demonstrated by the example). Since I have no idea about physics I do not want to touch the article, but I would be happy if some expert could clarify (and source) it. Brontosaurus (talk) 19:53, 26 February 2012 (UTC)

Four types of generating functions

Hi, I added a parenthetical about the existence of mixtures of the four types of generating functions, as it is an important caveat. I'm taking this from Goldstein's 3rd edition. It's on my p.374 in the paragraph above eqn. 9.20. "Finally, note that a suitable generating..." 137.99.19.141 (talk) 13:53, 20 July 2015 (UTC)

Form of an equation?

There's one thing missing here that I see asked across wikipedia: what is the form of an equation? The form of Hamilton's equations? How can we discuss form preservation here without first defining what form is? — Preceding unsigned comment added by Ponor (talk • contribs) 20:12, 3 August 2020 (UTC)

Error in Goldstein?

Equation 9.71 in Goldstein classical mechanics 3rd ed claims the Poisson matrix is given as $\{\varepsilon ,\varepsilon \}_{\eta }=M^{T}JM$ but expanding the matrix gives me Lagrange matrix instead. I have checked the conventions carefully and concluded that it must be an error. The equation immediately before eq 9.74 in the same book implies that it must be $MJM^{T}$. Two other sources, Lemos and Giacaglia from the wiki, also contradict Goldstein on this. For canonical transformations both matrices will equate to J anyway but I have made changes to use the proper matrix. EditingPencil (talk) 18:16, 25 December 2023 (UTC)