Talk:Poisson bracket

Disambiguation

Regarding the statement that "the smooth functions on a symplectic manifold are naturally a Poisson algebra" (paraphrase): I changed the link from Symplectic space to Symplectic manifold since manifolds are the objects with smooth functions, and 'symplectic space' refers (at least in WP) to either symplectic vector space or symplectic manifold. The natural domain of the "smooth functions on" functor is the category of manifolds, so it seems to be unambiguous. Sympleko 19:07, 19 Apr 2005 (UTC)

Additions?

Perhaps a discussion of non-constant Poisson structures (or brackets) would be helpful. The page on Moyal products mentioned the condition "constant Poisson structure" (paraphrase), and it seems like this page would be the right one to describe what that means. arichar6 6 Sep 2005

a link to poisson algebras in the "see also" section

Why is there a link to "superPoisson algebra"-s but not to "Poisson algebra"-s? I don't quite understand how the former are closer related.

Poisson algebra is linked from the second sentence at the top of the article, so does not need a "see also" link. -lethe ^{talk +} 04:26, 11 July 2006 (UTC)

General definition

The general definition given here, while correct, seems to be overly verbose. I believe that one can much more simply define the Poisson bracket by noting that if the symplectic form is nondegenerate, then it has an inverse tensor w^-1. Then one defines {f,g}:=w^-1(df, dg) (see for example the PlanetMath site, which has this much simpler description). This is much simpler than the lengthy definition given, although the details discussed there are perhaps sufficiently important to warrant keeping them. However, perhaps we should put this shorter version in as well? Mad2Physicist (talk) 08:05, 25 November 2008 (UTC)

Please, this index notation indeed is disgusting. I found a nice commentary in the german wiki for Poisson brackets: It runs like that: pfffttt the dust of 200 years. 200 years ago, Poisson would have gotten the Nobel prize for mathematics for his Poisson brackets. But today? Poisson brackets should be defined in a first approach on symplectic vector spaces with nothing else than the symplectic structure and the directional derivative, according to J-M Souriau p. 86. Then they could be generalized to symplectic manifolds. If someone doesn't believe that this approach is much easier to handle than the index-loaded one here, go to the website http://www.earningcharts.net/ipm/ipmWeylP.htm and calculate the examples, it is easy. — Preceding unsigned comment added by 130.133.134.35 (talk) 19:46, 5 October 2011 (UTC)

There was a sign error in the derivation of Hamilton's equations, so that a Poisson bracket which should have been {f,H} was instead written as {H,f}. I fixed it. Idempotent (talk) 15:40, 15 February 2012 (UTC)

I rewrote this section to simplify and clarify it. A bit of the material on mutual involution was moved to the constants of motion section. The material on the bivector was dropped because it is covered under the entry on Poisson manifold. I eliminated co-closed in favor of the more standard "symplectic" and co-exact in favor of "Hamiltonian." I gave the simpler definition up front and I think I made the logical flow more apparent. I also tried to connect it to the coordinate language of the earlier sections.

Finally, I changed the title, since "Definition" didn't make much sense to me, since there had already been a definition, and this section not only defines it but develops its basic properties.

Natkuhn (talk) 05:00, 24 December 2012 (UTC)

Lie algebra

Much of the intro paragraph here was covered in the previous section. The statement that the Lie algebra of functions under the PB is the Lie algebra of the group of symplectomorphisms is, I believe, incorrect--it is the Lie algebra of symplectic vector fields which is. This is the same as functions MODULO locally constant functions... As a result, I changed the title. Finally, I changed "vector field on the tangent bundle" to "vector field on the configuration space."

Frankly, I wasn't sure about the importance of this result and whether it merited inclusion in the article, but I didn't want to just remove the section. But as far as I'm concerned, it could be deleted. Natkuhn (talk) 05:09, 24 December 2012 (UTC)

Introduction

May i suggest that the current introduction is unintelligible to most people? Could someone rephrase it to introduce the basic concept in less technical terms?Richwil (talk) 18:30, 21 January 2009 (UTC)

Poisson Brackets are a deeply technical device arising out of a deeply theoretical portion of classical mechanics. This isn't the sort of thing which can be made more intelligible without distorting the concept. —Preceding unsigned comment added by 128.151.144.57 (talk) 20:37, 3 May 2009 (UTC)

No Richwil, this is impossible. If you want to understand mathematics, you have to learn mathematics, in the same way as you - some time ago - learned your language. The only question is whether the concept really needs such an old fashioned notation. — Preceding unsigned comment added by 130.133.134.16 (talk) 08:28, 6 October 2011 (UTC)

Animated Graphics

If nobody objects then I shall remove the animated graphics. The graphics only illustrates trivial and unrelated matter at an inappropriate level. It doesn't explain anything. The animation only catches the eye and makes it difficult to read the text. —Preceding unsigned comment added by Rdengler (talk • contribs) 07:23, 16 May 2009 (UTC)

Field Theoretical Poisson Bracket

There is field theoretical Poisson bracket expressed as an integral instead of a sum, and the partial derivatives become functional derivatives. It deserves at least a section in this article, or perhaps its own article. It would be nice if somebody feels a sudden urge to write it. A reference could be Walter Greiner, "Field Quantization". YohanN7 (talk) 21:35, 7 April 2013 (UTC)

Landau's definition

Landau defines the bracket as the minus of the definition given here. I haven't seen any other author do that, but perhaps it should be noted near the reference, in order to avoid confusion?

Explain, please!

Another article written by some expert who can't relate his knowledge to the world of non-experts. Wikipedia is getting worse every day! — Preceding unsigned comment added by Koitus~nlwiki (talk • contribs) 13:41, 3 August 2020 (UTC)

This material is inherently more advanced. Recommended material to learn first is Hamiltonian systems, some examples, Lie derivative, Liouville's theorem, symplectic form.

194.230.148.166 (talk) 21:11, 24 October 2022 (UTC)

Generating new invariants via the Poisson bracket

"Furthermore, according to Poisson's Theorem, if two quantities and are explicitly time independent () constants of motion, so is their Poisson bracket . This does not always supply a useful result, however, since the number of possible constants of motion is limited ( for a system with degrees of freedom), and so the result may be trivial (a constant, or a function of A and B.)"

But isn't this method quite productive in infinite dimensions?

194.230.148.166 (talk) 21:08, 24 October 2022 (UTC)

Is this a mistake in the post?

The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems 173.68.173.92 (talk) 04:37, 10 August 2023 (UTC)