Suspension of the billiard maps in the Lazutkin's coordinate

In this paper we proved that under the Lazutkin's coordinate, the billiard map can be interpolated by a time-1 flow of a Hamiltonian \begin{document}$ H(x,p,t) $\end{document} which can be formally expressed by \begin{document}$H(x,p,t)=p^{3/2}+p^{5/2}V(x,p^{1/2},t),\;\;(x,p,t)∈\mathbb{T}×[0,+∞)×\mathbb{T},$ \end{document} where \begin{document}$ V(·,·,·) $\end{document} is \begin{document}$ C^{r-5} $\end{document} smooth if the convex billiard boundary is \begin{document}$ C^r $\end{document} smooth. Benefit from this suspension we can construct transitive trajectories between two adjacent caustics under a variational framework.