That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian *Ht* on *D2* such that the corresponding flow at time 1 is equal to $\psi$.


I found this in claim a paper, and I think it should be easy, but nothing comes to mind. I'd be happy with a reference to a page in McDuff-Salomon, but I couldn't find this there immediately.


Thanks!



Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?
It is a theorem of Smale that the group of orientation-preserving diffeomorphisms of $D^2$, rel boundary, is contractible. If the diffeomorphisms can move the boundary, you can establish a homotopy equivalence between that and the circle. The diffeomorphisms do not have to preserve area. Then, a [theorem of Moser](https://mathoverflow.net/questions/7817/normal-coordinates-for-a-manifold-with-volume-form) establishes a deformation retract from diffeomorphisms to volume-preserving diffeomorphisms. Moser's result is easier to see if you have a closed manifold, but it extends to manifolds with boundary with the doubling trick. Together, this indirectly gives you a curve of symplectomorphisms connecting the identity to $\psi$, since in two dimensions the symplectic structure is just a volume structure. Finally if you have a smooth curve of area-preserving diffeomorphisms of a disk, I think there is a time-dependent Hamiltonian obtained by integrating the corresponding vector field.




---


I shared the same concern that Ilya expresses in the comments, but after considering it, here is why I think that it works. To have a clean view of the boundary conditions, let's double the disk to the sphere and let everything be equivariant with respect to reflection across the equator.


Moser's theorem truly is a deformation retraction. Let $M$ be a Riemannian manifold, let $\mu$ be a volume form on $M$ (not necessarily Riemannian volume), and let $\phi\_\alpha:M \to M$ be a family of diffeomorphisms of $M$ that may or may not preserve $\mu$. Then $\mu\_\alpha = (\phi\_\alpha)\_\*(\mu)$ is "wrong". Let $\mu\_{\alpha,t}$ be a family of volume forms defined as the weighted geometric mean of $\mu\_\alpha$ and $\mu$:
$$\mu\_{\alpha,t} = \mu^t \mu\_\alpha^{1-t}.$$
Then there is a corresponding Moser flow $\phi\_{\alpha,t}$ such that $\phi\_{\alpha,0} = \phi\_\alpha$ and $\phi\_{\alpha,1}$ is volume-preserving. Moreover, $\phi\_{\alpha,t} = \phi\_\alpha$ for all $t$ if $\phi\_\alpha$ is already volume-preserving for some fixed $\alpha$.


In particular, if $\phi\_t$ is a curve of diffeomorphisms as produced by Smale's theorem with $\phi\_0$ the identity, then Moser gives you an improvement $\phi\_{t,s}$ such that $\phi\_{t,1}$ is then what you want. What worried us is whether $\phi\_{1,1} = \phi\_1$; if $\phi\_1$ is area-preserving, then it is true.


