Talk:Symplectic manifold

Expansion

The following is an excerpt from the last version of this page which I thought would be better moved to the talk page. -- Fropuff 06:27, 2004 Feb 24 (UTC)

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The relation between symplectic geometry and Hamiltonian mechanics should explained in more detail.

Symplectic capcities should be mentioned.

Examples would be nice. Why are these things studied? I suspect because of physics?

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Moved this here from the article, and wikified it, for more convenient discussion:

Note that Emil Artin (founder of a german school of mathematicians in the Bourbaki style) already very early contributed to symplectic structures, which should be given in a more modern version than that at the top of this article. A symplectic vector space is a real vectorspace with a nondegenerate bilinear form, which is skew. This implies even-dimensionality. There are implications for multilinear algebra, (Weyl algebras, as opposed to Clifford algebras, where the bilinear form is symmetric), differential geometry (symplectic manifolds) and Physics (Poisson brackets in classical mechanics, canonical commutation relations in quantum mechanics). Look at http://www.EarningCharts.NET/ipm/ipmSympl.htm for more information. Hannes Tilgner

So, Artin was actually Austrian ... he wrote on geometric algebra in a quite broad sense. How much of this belongs here? User:Fropuff and I have already chewed the fat about linear symplectic space = symplectic vector space. Symplectic manifold could contain allusions to many of the mentioned things. Perhaps a bigger Related Articles section?

Charles Matthews 08:46, 5 May 2004 (UTC)

Hamiltonian Mechanics

We are told (in the section on Hamiltonian Mechanics IIRC) that it is possible to reach Hamiltonian mechanics from symplectic spaces directly, without going through Lagrangian methods.

This article (and several others in related areas of symplectic topology) does not indicate (to me!) how this is possible - at some stage a physicist needs to see some correspondence with forces and particles however abstract, since at the last Newton's Laws or their generalisation are experimental.

Bob aka Linuxlad 14:01, 9 Nov 2004 (UTC)

Basically with the background 2-form in place, the Hamiltonian H gives rise to a vector field and so the dynamics. Not really done with mirrors - the geometry is relatively simple. There is a sense in which the Lagrangian and Hamiltonian approaches are dual (not that I'd want to be pinned down, but the words Legendre transformation come to mind). So there ought to be an alternate way of looking at it. I suppose, roughly speaking, it's whether conservation of energy is expressed by a family of contours, or a family of orthogonal contours which expresses how your toboggan goes downhill. Charles Matthews 16:00, 9 Nov 2004 (UTC)

nondegenerate

For non-experts it might not be clear that a 2-form is nondegenerate iff it is seen as a bilinear form $\omega _{x}:T_{x}M\times T_{x}M\to \mathbb {R}$ pointwise. The reference explains only nondegeneracy in the linear setting. Hottiger 22:29, 14 April 2006 (UTC)

Is this why it is antisymmetric? The definition employs skew symmetry without transparently requiring skew symmetry, it would be great if that could be fixed. 150.203.48.23 (talk) 05:19, 10 September 2008 (UTC)

The requirement appears in the fact that a symplectic structure is a differential 2-form, and these are always antisymmetric. Orthografer (talk) 12:53, 10 September 2008 (UTC)

Why is that part of the definition not simply omitted? --pred (talk) 23:09, 3 December 2009 (UTC)

If anybody doesn't understand the article but is familiar with black holes consider this:

Isn't a sympletic form a 'sideways metric tensor'. The metric tensor is 'expected' to return 1, but the sympletic form is 'expected' to return 0. Everything else follows. It is quite like cos(theta), but instead one uses sin(theta) like magic. Extending that properly leads to everything described here except what is 'decided' but not realistic in crazy cases, for example a non-global metric is crazy but charged particles curving opposite ways might show that to us. The metric is assumed global to be a everyday metric, but ships are not made to stay in the harbor. I would use common sense and judge for myself what constraints are needed. The idea or principle is what matters, and everything around me is subject to changes depending on the situation. That is why many modern definitions are absolutely terrible. The situation determines the constraints NOT the mathematicians. But it is a good read, it sets the default and extends 'sin(theta)' well like the metric extends 'cos(theta)' well. Ironically when things get REALLY messy, One doesn't need much else and things get simpler. The less messy situations often are the most complex ones because they are too 'trendy'. In the future I would bet the equations and definitions are MUCH simpler than they are today because we are NOT god, so we cannot truly decide constraints. Definitions that decide constraints are terrible, and incomplete definitions are the path forward for my work. Ideally No definition should be complete. Leave that to the infinite future. The constraints that the sympletic form is closed may one day be completely violated by some kind of weapon so I wouldn't believe that in practice. Proof of principle and the idea of a sympletic form is incredible, but I won't assume anything much. I could loosely say that the metric is 1(1,1), and the sympletic form is i(1,1), and leave it at that. Unitary operators are very happy here. The metric should define vv/(vXv) and the sympletic form should define ivv/(vXv) whatever v and i are, and we really don't know. I would think the BEST way to define this, is to redefine cos(theta) and sin(theta) unifying the metric and sympletic form. The complete unification would then redefine e^(i*theta), right? Of course guess what that resembles?! That is a kahler metric! Yay! One simply needs a quadratic order version of some_crazy_trig_function(theta) in multiple dimensions and it seems the equations here would work nicely. For the sympletic form sin(theta) is redefined, and that (through some crazy extended dot product viewpoint) gives some of the equations and principle in the article without too many assumptions. The basic analogy that matches all this stuff is Met->cos(theta),Sympletic_Form->sin(theta). 2603:7080:8C03:B902:15D7:22F4:D4C2:221A (talk) 14:55, 17 October 2025 (UTC)

Kahler manifold

The sentence "A symplectic manifold endowed with a metric that is compatible with the symplectic form is a Kähler manifold" in the "Special cases and generalizations" section is not exactly correct; a Kahler manifold requires that the almost complex structure obtained from the metric and symplectic form be integrable. This should be stated somewhere, somehow, or Kahler should be changed to almost-Kahler. What sounds like the best course of action? 136.152.180.51 23:16, 19 April 2007 (UTC)

Hope this satisfies. Orthografer 06:30, 20 April 2007 (UTC)

WikiProject class rating

This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 10:04, 10 November 2007 (UTC)

Recent additive descriptions

I appreciate the recent descriptions, in particular, about motivation. Thanks!--Enyokoyama (talk) 11:31, 24 March 2013 (UTC)

A symbol needs to be defined

In the article section titled Motivation, I see the following sentence:

"So we require a linear map TM → T*M, or equivalently, an element of T*M ⊗ T*M."

The (operator?) T is not defined or described. Is that the symplectic 2-form alluded to above? I'm hoping someone can prepend something like, "Given a real manifold M and a symplectic form T..." (or whatever T actually is!)

Thank you. Dratman (talk) 05:31, 14 March 2017 (UTC)

Done. Its the tangent manifold. 172.58.187.254 (talk) 01:07, 15 August 2021 (UTC)

Another definition of isotropic

I have seen elsewhere a riemannian manifold being called isotropic if there is an isometry that rotates an arbitrary angle around each point. If anyone finds a source, it would be nice to add a disambiguation to distinguish the two meanings. Madhing (talk) 17:01, 15 September 2024 (UTC)