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 \[ H(x,p,t)=p^{3/2}+p^{5/2}V(x,p^{1/2},t),\quad(x,p,t)\in\T\times[0,+\infty)\times\T, \] 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.