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 $H(x,p,t)$ which can be formally expressed by $$H(x,p,t)=p^{3\mathrm{/}2}+p^{5\mathrm{/}2}V(x,p^{1\mathrm{/}2},t),(x,p,t)\in \text{T}\times [0,+\mathrm{\infty })\times \text{T},$$ where $V(\cdot ,\cdot ,\cdot )$ is $C^{r-5}$ smooth if the convex billiard boundary is $C^r$ smooth. Benefit from this suspension we can construct transitive trajectories between two adjacent caustics under a variational framework.