# A variational approach to symplectomorphisms

Let $(\Sigma,\omega)$ be a compact symplectic surface, and let $\mathcal{D}$ be the group of diffeomorphisms of $\Sigma$. Is there some known variational approach to determine if an element $f\in\mathcal{D}$ is a symplectomorphism? I.e. does there exist a functional $$F:\mathcal{D}\longrightarrow\mathbb{R}$$ such that the minima of $F$ are exactly the symplectomorphisms?

Would the following functional work? $$F(\phi) = \int_\Sigma(1-\det(d\phi))^2\omega,$$ where $\det(d\phi)$ is the smooth map defined by $\phi^*\omega = \det(d\phi)\omega$. Then $F(\phi) = 0$ if, and only if $\phi^*\omega = \omega$, and this is an extremum in the path connected (i.e. isotopy) components of $\mathcal{D}$. Can it be analyzed in more depth somehow?

• If $\phi^*\omega=\omega$, then $\det(d\phi)$ is identically $1$ according to your defintion, and $F(\phi)$ is the volume of $\Sigma$, not zero. – Mariano Suárez-Álvarez Jun 26 '15 at 8:29
• @MarianoSuárez-Alvarez You're right. I've corrected the definition of $F$. – Daniel Robert-Nicoud Jun 26 '15 at 8:41
• Take a look at Meiss' Symplectic maps, variational principles, and transport, it might have the material that you are interested in. You can find it here: amath.colorado.edu/faculty/jdm/papers/SymplecticMapsReview.pdf – Evgeny Jun 27 '15 at 11:52