Talk:Momentum map

message: I see that the PlanetMath page 'momentum map' rather suggests this should be tied in with the coadjoint action. Charles Matthews 21:32, 13 January 2006 (UTC)
thread_title: Untitled

message: My understanding is that the moment map is required to be equivariant with respect to the action of the group on the manifold and the coadjoint action of the group on the dual of its Lie algebra. Yggdrasil014 23:20, 22 January 2007 (UTC)
replies: text: Yes: this is a translation of the fact that the map from the Lie algebra of G to the Lie algebra of functions under Poisson bracket is required to be a Lie algebra homomorphism. Geometry guy 10:21, 10 February 2007 (UTC)
thread_title:  equivariance 

message: Which title would be preferred? As I understand it, geometers tend to use moment map, whereas physicists are more likely to say momentum map. I'm a geometer, but I always say "momentum map", because I believe the term "moment map" is a mistranslation of the term application moment coined in Souriau's original (French) article. The terms "moment" and "momentum" are not synonymous in English, and "momentum" is the relevant word in this context.
Geometry guy 10:31, 10 February 2007 (UTC)
replies: text: Thanks for pointing out the possible French origin. I don't care much about which we use --- especially since Momentum map already redirects here --- but there are a few points to keep in mind. Original research is not allowed; the terminology, the history, and the math itself should all be backed up with sources. We should be using whatever terminology is commonly used, whether its etymology is simple or not. Perhaps you could add a note about the etymology, if you can source it. My original choice of Moment map for the title arose from its use in Donaldson & Kronheimer and McDuff & Salamon, two standard references in geometry. But I'm not wedded to it. Joshua R. Davis 15:00, 10 February 2007 (UTC)
replies: text:  Good point. At the moment (excuse the pun), this article reads primarily like a symplectic geometry article (even though it is in wiki-project physics), in which case, I think "moment map" is probably still the prevalent term, although more and more geometers are using "momentum map". I'll add a reference to Souriau anyway. I'm happy to postpone the renaming issue: to some extent it depends on whether there is more physics input. Thanks for starting this article, by the way: it is a very important topic! Geometry guy 15:49, 10 February 2007 (UTC)
replies: text: Now it's part of WikiProject:Mathematics as well. I labeled it "geometry" rather than "mathematical physics", because it enjoys geometric application independent of physics. Joshua R. Davis 15:47, 12 February 2007 (UTC)

text:  Great! And I agree with your choice. Still, I think it is worthwhile to attract physicists to this article.... Geometry guy 23:37, 12 February 2007 (UTC)
thread_title:  Moment vs momentum 

message: I've changed the Math importance rating from low to mid. The moment map is far from "peripheral knowledge, possibly trivial"; it is a key tool in symplectic geometry and mathematical physics, and ties in to toric geometry as well --- which reminds me that we need to talk about that here. Joshua R. Davis 15:26, 1 March 2007 (UTC)
replies: text: I fully support the uprating. Geometry guy 17:31, 1 March 2007 (UTC)
thread_title: Importance

message: 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 10:00, 10 November 2007 (UTC)
thread_title:  WikiProject class rating

message: One of the most beautiful properties of the moment(um) map is the convexity of its image (Atiyah, Guillemin-Sternberg, Kirwan theorems).  I think this should be included in the article. Perhaps I'll have a go at writing something, unless someone else does it first. Simplifix (talk) 15:16, 12 February 2008 (UTC)
replies: text: I agree that it should be discussed. Please do have a go. Joshua R. Davis (talk) 15:49, 12 February 2008 (UTC)
thread_title:  Convexity 

message: In the "formal definition" section, wouldn't it be simpler to put 
:\, (R_x)_* \xi, 
where R_x is right translation by x, instead of 
:\left.\frac{d}{dt}\right|_{t = 0} \exp(t \xi) \cdot x? 
I don't want to change it myself because I am not very familiar with this area.  —Preceding unsigned comment added by 24.30.147.224 (talk) 02:25, 2 October 2008 (UTC)
replies: text: x is an element in the manifold so R_x does not make sense (the manifold is not necessarily a Lie group).Eok20 (talk) 23:05, 17 May 2010 (UTC)
thread_title:  exponential map necessary? 

message: Ana Cannas da Silva's book Lec. on sym. geo. showed that(atribute to Atiyah & Bott) the space of connection on a principal bundle is a (infinite dimension) symplectic manifold, the group of gauge tran. act on this space Hamitonian, the curvature is the moment map!--刻意(Kèyì) 06:12, 23 March 2009 (UTC)
thread_title: Curvature is a moment map

message: Substituted at 00:20, 30 April 2016 (UTC)
thread_title: Assessment comment

message: The definition of momentum map as stated uses exactness, not closedness. Panpach 11:39, 2. October 2018 (UTC)
:The text was correct, but I have added text to clarify the relationship. Mgnbar (talk) 17:52, 2 October 2018 (UTC)
replies: text: Unfortunately, the new text is redundant with the subsequent section on Hamiltonian actions. Mgnbar (talk) 20:16, 2 October 2018 (UTC)
thread_title: Closed vs. exact