Talk:Symplectic manifold
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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)
- 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?
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)
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)
[edit] nondegenerate
For non-experts it might not be clear that a 2-form is nondegenerate iff it is seen as a bilinear form 
pointwise. The reference explains only nondegeneracy in the linear setting. Hottiger 22:29, 14 April 2006 (UTC)
