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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 3rd 2012
• (edited Jul 3rd 2012)

I wrote Hamiltonian action.

I tried to say precisely what the action is by. In the literature (but also in a previous version of our moment map entry) there is often (for instance on Wikipedia, but also in many other sources) an imprecise (not to say: wrong) statement, where an action by Hamiltonian vector fields is not distinguished from one by Hamiltonians.

• CommentRowNumber2.
• CommentAuthorjim_stasheff
• CommentTimeJul 3rd 2012
You might want to add to
Dualizing, the homomorphism μ is equivalently a linear map
μ˜:X→
• CommentRowNumber3.
• CommentAuthorjim_stasheff
• CommentTimeJul 3rd 2012
uh oh, the rest didn't appear

which is a homomorphism of Poisson manifolds. This is called the moment map of the (infinitesimal) Hamiltonian G-action.

You might want to add that 1st class constraints give something like mu but with range just a vector space and no G in sight.
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 3rd 2012

Thanks, good point.

I made a quick preliminary note at moment map - Relation to constrained mechanics.

Currently symplectic reduction points to BRST-BV formalism and much stuff is hidden there which eventually ought to get its own entries. Maybe later…

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeJul 4th 2012

You have both $HamSympl(X, \omega)$ and $Sympl(X, \omega)$ for Hamiltonian symplectomorphisms. Which is better?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJul 4th 2012

The second is supposed to be just for symplectomorphisms, not necessarily Hamiltonian. If I mixed that up, then it needs to be corrected. I can’t check right now, (am at a bus stop, and there is my bus!…)

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeJul 4th 2012

So it looks like the $Sympl(X, \omega)$ needs a ’Ham’ in front of it. I’ll do that.

Also, I see that the links from ’Hamiltonian symplectomorphism’ points to symplectomorphism, and there’s nothing there to explain the difference between ordinary ones and Hamiltonian ones.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 4th 2012

Thanks, David! You are quite right, there was lacking a bunch of information. I have now added in more lines at Auto-symplectomorphisms.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeJul 4th 2012
• (edited Jul 4th 2012)

There’s also a notion of Hamiltonian diffeomorphism, e.g., here. Do we need $HamDiff$? But maybe that’s your $HamSymp$.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJul 4th 2012

Yes, that’s just another term for the same concept. Some people also say “symplectic diffeomorphism” instead of “symplectomorphism”, e.g. here. I have added in this alternative terminology to the entry symplectomorphism now.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeJul 4th 2012

So then, should we include the direct definition of Hamiltonian diffeomorphism/symplectomorphism from here:

A symplectomorphism $\phi \in$ Symp(M, $\omega$) is said to be a hamiltonian diffeomorphism if there exists a hamiltonian isotopy $h_t$ such that $\phi = h_1$.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJul 4th 2012
• (edited Jul 4th 2012)

Okay, I have expanded

The further subgroup corresponding to those symplectic vector fields which are Hamiltonian vector fields […]

to

The further subgroup corresponding to those symplectic vector fields which are flows of Hamiltonian vector fields coming from a smooth family of Hamiltonians […]

I don’t feel like expanding further on this at the moment. But of course if you or somebody feels like adding more from the literature, please do! Eventually then we could also add formal Definition / Proposition environments to the entry.