Difference in usual contact structure on $\mathbb R^3$ I have seen the usual contact structure as either the kernel of $dz + xdy$ or the kernel of $dz - ydx$. They are contactomorphic, of course, just rotated. 
However, I was wondering if there are any particular reasons why one would use one form over the other as the ``standard''? I had googled it to no avail. I did find that Etnyre states that "many authors prefer to use" the latter, but with no reasons named.
 A: Historically, contact transformations were introduced by S. Lie in the late nineteenth century in his attempts to get a general understanding of ordinary differential equations of order $1$, namely, problems having the following form $F(x,y,y')=0$. To do so, he introduced the notion of line element (which generalizes as contact elements for arbitrary smooth manifolds and lead to the study of contact geometry!), it consists of a point $(x,z)\in\mathbb{R}^2$ and a nonvertical line $\ell$ of $\mathbb{R}^2$ passing through it, such a line is of the form:
$$\mathrm{d}z-p\,\mathrm{d}x=0,$$
where $(x,z)$ is taken as the origin of $\mathbb{R}^2$, $(\mathrm{d}x,\mathrm{d}z)$ is the dual basis of $\mathbb{R}^2$ and $p$ is the slope of $\ell$. The set of all line elements of $\mathbb{R}^2$, denoted by $C\mathbb{R}^2$, can be identified with $\mathbb{R}^3$ with coordinates $(x,z,p)$ and is naturally endowed with the globally defined contact form $\alpha_0\colon(x,z,p)\mapsto \mathrm{d}z-p\,\mathrm{d}x$. Using this setting, solutions to differential equations of order $1$, now, correspond to integral curves of the plane distribution $\ker(\alpha_0)$, in other word they are legendrian curves of $C\mathbb{R}^2$! 
The theory can be developed further introducing $X_F$, the Lie characteristic vector field of $F(x,y,y')=0$, its integral curves are exactly the solutions of the given ODE, provided that $\alpha_0$ and $\mathrm{d}F$ are linearly independent along the level set $\{F=0\}$.
To wrap things up, I see mainly two reasons to have a preference for $\mathrm{d}z-p\mathrm{d}x$:


*

*It was the one historically studied and emerges in a very natural setting.

*It provides a self-contained and concise theory of differential equations of order $1$ and one may even argue that it contains in germs the method of characteristics for partial differential equations.
