Talk:Symplectic vector space

I redirected this page to symplectic manifold (which symplectic topology and symplectic geometry also go to) and then upon further reflection I reverted the page to the way it was. Shouldn't this page discuss a symplectic vector space and not a symplectic manifold (as it does now). Briefly a symplectic vector space is a vector space with a nondegenerate, alternating form. The tangent space to every symplectic manifold is a symplectic vector space. Actually, the example given on this page really is just a vector space. -- Fropuff 07:03, 2004 Feb 24 (UTC)

The article is not complete but is probably not a stub anymore. Should that line be deleted? Sympleko 10:57, 13 Apr 2005 (UTC)

symplectic matrix

I'm completely befuddled by this recent edit:

(Note that this is not the same thing as a symplectic matrix, which is a different concept discussed below).

which was used to replace

An ordered basis can always be found to express this matrix as a symplectic matrix.

The new sentance seems to be trying to say that something isn't something, but I'm not sure what the two somethings are. The old sentance seemed quite clear to me: a skew symmetric matrix can always be made symplectic by change of coords ... right? why remove such a sentance ? Confused ... linas 03:43, 11 May 2005 (UTC)

Symplectic matrices are the coordinate representations of symplectic transformations on a symplectic vector space. They presevere the non-degenerate skew-symmetric matrix which is the coordinate representation of the symplectic form. Symplectic matrices are only defined with respect to said non-degenerate skew-symmetric matrix. To say that the symplectic form has a coordinate representation as a symplectic matrix doesn't really make any sense: with respect to which matrix? -- Fropuff 05:03, 2005 May 11 (UTC)

Yes, Fropuff is right. So let me add, that instead of writing on the Gram-Schmidt construction of a Darboux basis, it is better to include here the direct construction of a Darboux basis. It's not that difficult, it took me just twenty minutes, using the symplectic bilinear form and nothing else. So this construction belongs actually into the first section, without referring to a matrix representation at all. — Preceding unsigned comment added by 130.133.134.16 (talk) 12:39, 26 September 2011 (UTC)

Analogy with complex structures

I'm a bit confused as to why this section brings up manifolds and their tangent spaces. Would it perhaps be more direct to say that, as can be seen from the standard symplectic form above, every symplectic form on $\mathbb {R} ^{2n}$ is isomorphic to the imaginary part of the complex inner product on $\mathbb {C} ^{n}$ (with the convention of the first argument being anti-linear)? Adam Marsh (talk) 00:30, 13 March 2018 (UTC)

Why only define a symplectic space over the real numbers?

Shouldn't this article be far more general? It is completely possible to define symplectic spaces over arbitrary fields. It might be possible though, that those spaces only make since over fields with characeristic unequal to 2. See Quadratic_form. —Preceding unsigned comment added by 169.229.55.42 (talk • contribs) 01:26, 26 January 2006

In short, yes. The real case is the easiest one to handle, and is the one that is revelant to symplectic manifolds. In any case, I believe everything that is stated in the article works over arbitrary fields with char ≠ 2. I need to look up how the characteristic 2 case is handled. I suspect one just uses nondegenerate alternating forms rather than skew-symmetric ones, but there may be additional subtleties. -- Fropuff 02:08, 26 January 2006 (UTC)

Equal or just Isomorphic?

Shouldn't a Lagrangian subspace W of a symplectic vector space be merely isomorphic to W-perp, not equal? Sdenton-at-math.ucdavis 22:52, 16 February 2007 (UTC)

No. If that were the case any subspace whose dimension was half that of V would be Lagrangian, which is not a very useful notion. -- Fropuff 00:15, 17 February 2007 (UTC)

Symplectic group

Here one should note, that the symplectic group may be looked at as the automorphism group of the symplectic vector space and its Lie algebra, developped over the convergent exponential series for linear transformations is its derivation Lie algebra. Hans Tilgner

Symplectic Lie- and Jordan algebras

Yes, this Lie algebra should be included here, but also its Jordan-algebraic correspondence (of self-adjoint with respect to the symplectic form) linear transformations under the anti-commutator. — Preceding unsigned comment added by 130.133.134.35 (talk) 08:21, 14 November 2011 (UTC)

and the (global) structures which result from exponentiation, i.e. the symplectic group, a standard simple Lie group, and the set of invertible self-adjoint endomorphisms, a standard domain of positivity. Thus there result four categories, like for any bilinear form. I think that all four categories play a role in phase space and quantum mechanics. — Preceding unsigned comment added by 130.216.13.55 (talk) 20:54, 18 January 2014 (UTC)

Sugguestion

I strongly sugguest to use in the whole Wikipedia for the identity map on a set M the notation id_M (some authors use Id_M instead, but they are a minority). If the set happens to be the space of column vectors Fⁿ on a field F then the index should be the dimension of this column vector space. In this case id_n is the identity matrix. This notation would remove the difficulty that english speaking mathematicians mostly use I_n instead, german speaking E_n , ... . Hans Tilgner —Preceding unsigned comment added by 84.88.66.226 (talk) 21:31, 21 December 2010 (UTC)

in addition, for symplectic structures it makes sense to use the letters q and p instead of x and y. Same content, but physicists would prefer this (one even can think of lower indices for the q's and upper indices for the p's). — Preceding unsigned comment added by 130.216.13.55 (talk) 20:59, 18 January 2014 (UTC)

Heisenberg group

Here the letter K must be explained. It is the ground field K, I suppose. Hans Tilgner

Yes it is, I fixed it. 67.198.37.16 (talk) 18:25, 4 December 2020 (UTC)

Confusion in the symplectic map section

The section "Symplectic map" (in the article) seem to be confusing a symplectic vector space with a symplectic structure on a manifold. To my knowledge there is no reason to speak of the pullback, differential of linear map, and the tangent space of a vector space in the linear settings, only in the differential-geometry ones. Can someone comment? 132.64.72.151 (talk) 13:12, 14 July 2011 (UTC)

Does make sense to speak of pullbacks, does not make sense to talk of differentials. Or it may make sense, but is not relevant. I will remove that bit. -lethe ^{talk +} 05:10, 2 May 2012 (UTC)

Jean-Marie Souriau

Sorry, but the main source for symplectic structures is the book of J.-M. Souriau, which is available in French and English. And that of Claude Godbillon also. There you find the definition of a Poisson bracket, basis free! Note that coordinates have nothing to do with the problem. 184.22.189.33 (talk) 04:41, 18 February 2018 (UTC)

I assume you mean J.-M. Souriau (1997) "Structure of Dynamical Systems, A Symplectic View of Physics" Springer ? Looks awesome! I added it. I could not find the correct book by Godbillon, unless you mean this one: (1983) "Dynamical Systems on Surfaces" (Universitext) which seems too low-dimensional?? Or this one: Claude Godbillon (1969) "Géométrie différentielle et mécanique analytique", Hermann but it seems French-only. (I found a Persian translation!) 67.198.37.16 (talk) 18:32, 4 December 2020 (UTC)