  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.

Dynamical Systems