As the title suggests, are there any reference that deals with geodesics in the symplectisation $$\Bbb R \times X$$ for a contact manifold $(X,\alpha)$. We equip the symplectisation with the symplectic form $d(e^s\alpha)$ and with an $s$-invariant almost complex structure.
In the case $X$ is $S^1$ for example, then $\omega = d(e^sdz)$ where $z$ is the radial coordinate on the circle. The metric tensor is given (according to my computations) by the matrix $$\begin{matrix} e^s & 0 \\ 0 & e^s\end{matrix}$$
(w.r.t the coordinates $(s,z)$ representing the $\Bbb R$-coordinate and the radial coordinate on $S^1$). After computing the Christoffel symbols I think I can conclude that the paths $$\begin{cases} s(t)&=\log(t^2) \\ z(t) &= 0 \end{cases}$$ and $$\begin{cases} s(t)&=0 \\ z(t) &= t \end{cases}$$
Are geodesics. Are there analogous results in higher dimensions? I'm interested especially in showing that the Reeb vector field is geodesic and to study the "form" of the geodesics that travel the neck in the "$\Bbb R$" direction