Given *any* manifold $M$, we can get a *symplectic* manifold by taking the **cotangent bundle** $T^\ast M$ with symplectic form $\omega=\sum dp\_i\wedge dq\_i$. Given *any* manifold $M$, we can get a *contact* manifold by taking the **projectivization of the cotangent bundle** $\mathbb{P}^\ast M=(T^\ast M-\lbrace0\text{-section}\rbrace)/{\sim}$ where the contact form arises from the tautological 1-form on $T^\ast M$.


Given any *contact* manifold $(N,\lambda)$, we can get a *symplectic* manifold by **symplectization** $\mathbb{R}\times N$ with symplectic form $d(e^s\lambda)$. Continuing in the same spirit:


**Is there a** *"contactization"* **to pass from any given** *symplectic* **manifold to a** *contact* **one, making use of the symplectic data?**


Aside: I came across a paper of Eliashberg-Hofer-Salamon ([Lagrangian Intersections in Contact Geometry](http://www.math.ethz.ch/~salamon/PREPRINTS/ehs95.pdf)), and in certain scenarios we do indeed have one. If our symplectic manifold $M$ is *exact*, i.e. $\omega=d\alpha$, then $(M\times S^1,dz-\alpha)$ is a contact manifold. Now if we don't have exactness, there is at least a way to contactize $M$ when some positive multiple of $\omega$ represents an *integral* cohomology class in $H^2(M)$, and this is some principal $S^1$-bundle called ''pre-quantization''. **Is ''pre-quantization'' the only way to contactize here?**



'Contactization' and Symplectization
The "pre-quantization" construction of a contact manifold out of symplectic manifold predates prequantization by a couple of decades: 
see Boothby, W. M.; Wang, H. C.
On contact manifolds.
Ann. of Math. (2) 68 1958 721–734. 
The analogue of the theorem for symplectic orbifolds is due to Thomas: Thomas, C. B.
Almost regular contact manifolds.
J. Differential Geometry 11 (1976), no. 4, 521–533.


You may think of the Boothby-Wang construction as constructing a contact fiber bundle over a symplectic manifold with fiber $S^1$. If we look at the construction this way, it can be generalized. See my paper Contact fiber bundles.
J. Geom. Phys. 49 (2004), no. 1, 52–66. 


