most recent 30 from http://mathoverflow.net 2013-05-23T23:47:46Z http://mathoverflow.net/feeds/user/16813 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71571/example-of-symplectic-and-hamiltonian-diffeomorphism-on-s2-and-t2 Tatou Papora 2011-07-29T12:52:25Z 2011-07-31T11:20:20Z

<p>Hello everybody. For a purpose of consolidation of some result I am trying to set down, I need to construct an example to sustain the theory and I am looking for symplectic and Hamiltonian diffeomorphisms. So does someone can help me writing some non-trivial explicit examples of symplectic and Hamiltonian diffeomorphisms on compact surface? (at least examples for $S^2$ and $T^2$ ) NB : By surface I mean 2-dimensional manifold. Thanks a lot</p>

http://mathoverflow.net/questions/71571/example-of-symplectic-and-hamiltonian-diffeomorphism-on-s2-and-t2/71618#71618 Tatou Papora 2011-08-01T16:01:40Z 2011-08-01T16:01:40Z

The difference between hamiltonian diffeomorphism and symplectomorphism that is isotopic to the identity is quite small and describe by the flux homomorphism (The breakthrough by Banyaga). So any symplectic isotopy (path to identity in the group of symplectiomorphisms) with zero flux is ended by an Hamiltonian diffeo. Precisely speaking, the isotopy can be made homotopic to hamiltonian isotopy having the same ends.

http://mathoverflow.net/questions/71571/example-of-symplectic-and-hamiltonian-diffeomorphism-on-s2-and-t2/71583#71583 Tatou Papora 2011-08-01T15:52:57Z 2011-08-01T15:52:57Z

Sure. More than being homotopy equivalent it deformation retracts to $SO(3)$. One can find the proof in Mu-Tao's paper <a href="http://mrlonline.org/mrl/2001-008-005/2001-008-005-007.pdf" rel="nofollow">mrlonline.org/mrl/2001-008-005/&hellip;</a>. But the interst here is to handle an explicit diffeomorphism like one can write for example $f:\mathbb R\longrightarrow \mathbb R, x\mapsto \frac{x}{x^2+4}\sin(x)$ and whatever.

http://mathoverflow.net/questions/71571/example-of-symplectic-and-hamiltonian-diffeomorphism-on-s2-and-t2 Tatou Papora 2011-08-01T15:43:02Z 2011-08-01T15:43:02Z

Thanks all for your interest to my question. First of all, sorry for not being precise. So could use on $S^2$ $\sin\varphi d\theta\wedge d\varphi)$ in spherical coordinate or $d\theta\wedge d z$ in cylindrical coordinate. Feel free to consider the one that can be helpful. Besides, for $T^2$, $d\theta\wedge d\varphi$ where $\theta, \varphi \in S^1$ are general angular coordinates.