I think in priciple it's possible to consider a theory of "conformal-symplectic manifolds", in an analogous fashion as the usual conformal geometry. 


To spell out the spontaneous definitions: say that two symplectic forms $\omega\_1$, $\omega\_2$ on a smooth manifold $M$ are *conformal* to each other if there is a smooth positive function $\lambda \in \mathcal{C}^{\infty}(M,\mathbb{R}^{+})$ such that $\omega\_1=\lambda\cdot \omega\_2$ on $M$. Call a pair $(M,[\omega])$, with $[\omega]$ a conformal class of symplectic structures, a *conformal-symplectic* manifold. A smooth map $\varphi : M \to N$ between conformal-symplectic manifolds $(M,[\omega\_1])$ and $(N,[\omega\_2])$ is *conformal-symplectic* if $\varphi^\*(\omega\_2)\in [\omega\_1 ]$.


Just out of curiosity, I would like to ask:



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> Has such a theory been considered or studied? What can be said about these structures (provided it doesn't turn out to be somehow a "trivial" subject)? 
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Conformal-symplectic geometry ?
If the manifold has dimension bigger than 2, I think the conformal class of $\omega$ is just $k\omega$ for constants $k$. Locally, by Darboux we can write $\omega = \sum\_i dq^i \wedge dp\_i$. If the dimension is greater than 2, the only way for $0 = d(f\omega) = df \wedge \omega$ is if $df = 0$.


