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Suspension of the billiard maps in the Lazutkin's coordinate

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  • 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


    where $ V(·,·,·) $ 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.

    Mathematics Subject Classification: Primary:37J50;Secondary:70H08.


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  • Figure 1.  The reflective angle keeps equal to the incident angle for every rebound

    Figure 2.   

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