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I have expanded and edited moment map.
for completeness I have spelled out here the proof that the Lie-algebraic and the Poisson-algebraic definition of the moment map are indeed equivalent.
(This is straightforward unwinding of definitions. But somebody asked me and it seems in the literature this is always left as an exercise. So I spelled it out explicitly.)
added section References – In thermodynamics
Joe Onekun left the page badly broken
EDIT never mind, he fixed it as anonymous.
Change name to momentum map. Moment map is a misnomer and physically incorrect. It is an erroneous translation of the French notion application moment. See this mathoverflow question for the history of the name.
Tobias Diez
Thanks, but what one really should do is add any discussion (or pointer to it) of the history of the term to the entry.
Thanks. I have added the pointer also to the page carrying your name, here
Publication data for
And a previously missing coauthor
There are of course many technical terms that historically are misnomers and yet have become standard terminology (e.g. “imaginary numbers”).
You now made an entry on “momentum maps” almost all of whose references speak of “moment maps”.
I don’t have strong feeling about it, but I think better practice would be to go the other way: Leave the established terminology, so that people and search engines have an easy time finding the entry, and add a remark explaining what would have been more appropriate terminology.
Adding more references, all of which use “momentum map/mapping”.
Juan-Pablo Ortega, Tudor S. Ratiu, Momentum maps and Hamiltonian reduction, Progress in Mathematics 222 (2004). Birkhäuser Boston. ISBN 0-8176-4307-9.
Yvonne Choquet-Bruhat, Cécile DeWitt-Morette, Analysis, Manifolds and Physics, Elsevier, 1977. ISBN 978-0-7204-0494-4.
S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Oxford Science Publications, 1990. ISBN 0-19-850269-9.
Michèle Audin, Torus actions on symplectic manifolds, Progress in Mathematics 93 (2004), Birkhäuser, ISBN 3-7643-2176-8.
Karl-Hermann Neeb, Kähler geometry, momentum maps and convex sets, Tsinghua lectures in mathematics, 361–391. Adv. Lect. Math. (ALM), 45. International Press, Somerville, MA, 019. ISBN: 978-1-57146-372-2.
Re #45: I did a search on MathSciNet, there are 1905 and 4128 entries mentioning “momentum map/mapping” and “moment map/mapping”, respectively. I added several books that use “momentum map/mapping” to give a more balanced picture of the literature.
So I would say that both terms are well-established, and we should certainly prominently mention both of them. I think it’s okay for the nLab to use the term used by a third of all publications as the primary term, given the context.
Just as an aside, the nLab seems to have much more extreme examples of terminology mismatch, e.g., the article dualizing object uses as the primary name terminology that not only is not used by the overwhelming majority of publications, but is also in direct conflict with other uses of the same terminology, e.g., in star-autonomous categories.
Mention both “momentum” and “moment” in the first paragraph:
In symplectic geometry a momentum map (French: application moment, often erroneously translated as moment map) is a dual incarnation of a Hamiltonian action of a Lie group (or Lie algebra) on a symplectic manifold.
Fixed a reference:
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