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April  2017, 37(4): 2227-2242. doi: 10.3934/dcds.2017096

Suspension of the billiard maps in the Lazutkin's coordinate

Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, M5S 2E4, Canada

Received  July 2016 Revised  September 2016 Published  December 2016

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
$H(x,p,t)=p^{3/2}+p^{5/2}V(x,p^{1/2},t),\;\;(x,p,t)∈\mathbb{T}×[0,+∞)×\mathbb{T},$
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.
Citation: Jianlu Zhang. Suspension of the billiard maps in the Lazutkin's coordinate. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2227-2242. doi: 10.3934/dcds.2017096
References:
[1]

P. Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier, Grenoble, 52 (2002), 1533-1568.  doi: 10.5802/aif.1924.  Google Scholar

[2]

C.-Q. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geometry, 67 (2004), 457-517.   Google Scholar

[3]

C.-Q. Cheng and J. Yan, Arnold diffusion in hamiltonian systems: A priori unstable case, J. Differential Geometry, 82 (2009), 229-277.   Google Scholar

[4]

R. Douady, Une démonstration directe de lé quivalence des théorémes de tores invariants pour difféomorphismes et champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math., 295 (1982), 201-204.   Google Scholar

[5]

C. Golé, Symplectic Twist Maps: Global Variational Techniques, Advanced Series in Nonlinear Dynamics, World Scientific Pub Co Inc, 2001. doi: 10.1142/9789812810762.  Google Scholar

[6]

V. Lazutkin, The existence of caustics for a billiard problem in a convex domain, Izv. Akad. Nauk SSSR, SerMat. Tom, 37 (1973), 186-216.   Google Scholar

[7]

R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9 (1996), 273-310.  doi: 10.1088/0951-7715/9/2/002.  Google Scholar

[8]

J. Mather, Glancing billiards, Ergod. Th. Dyn. Sys., 2 (1982), 397-403.  doi: 10.1017/S0143385700001681.  Google Scholar

[9]

J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Mathematische Zeitschrift, 207 (1991), 169-207.  doi: 10.1007/BF02571383.  Google Scholar

[10]

J. Mather, Variational construction of connecting orbits, Annales de l'institut Fourier, 43 (1993), 1349-1386.  doi: 10.5802/aif.1377.  Google Scholar

[11]

J. Moser, Monotone twist mappings and the calculs of variations, Ergodic Theory and Dyn. Syst., 6 (1986), 401-413.  doi: 10.1017/S0143385700003588.  Google Scholar

[12]

J. Moser, Selected Chapters in the Calculus of Variations, Lectures in Mathematics. ETH Zürich, Birkhäuser, 2003. doi: 10.1007/978-3-0348-8057-2.  Google Scholar

show all references

References:
[1]

P. Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier, Grenoble, 52 (2002), 1533-1568.  doi: 10.5802/aif.1924.  Google Scholar

[2]

C.-Q. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geometry, 67 (2004), 457-517.   Google Scholar

[3]

C.-Q. Cheng and J. Yan, Arnold diffusion in hamiltonian systems: A priori unstable case, J. Differential Geometry, 82 (2009), 229-277.   Google Scholar

[4]

R. Douady, Une démonstration directe de lé quivalence des théorémes de tores invariants pour difféomorphismes et champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math., 295 (1982), 201-204.   Google Scholar

[5]

C. Golé, Symplectic Twist Maps: Global Variational Techniques, Advanced Series in Nonlinear Dynamics, World Scientific Pub Co Inc, 2001. doi: 10.1142/9789812810762.  Google Scholar

[6]

V. Lazutkin, The existence of caustics for a billiard problem in a convex domain, Izv. Akad. Nauk SSSR, SerMat. Tom, 37 (1973), 186-216.   Google Scholar

[7]

R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9 (1996), 273-310.  doi: 10.1088/0951-7715/9/2/002.  Google Scholar

[8]

J. Mather, Glancing billiards, Ergod. Th. Dyn. Sys., 2 (1982), 397-403.  doi: 10.1017/S0143385700001681.  Google Scholar

[9]

J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Mathematische Zeitschrift, 207 (1991), 169-207.  doi: 10.1007/BF02571383.  Google Scholar

[10]

J. Mather, Variational construction of connecting orbits, Annales de l'institut Fourier, 43 (1993), 1349-1386.  doi: 10.5802/aif.1377.  Google Scholar

[11]

J. Moser, Monotone twist mappings and the calculs of variations, Ergodic Theory and Dyn. Syst., 6 (1986), 401-413.  doi: 10.1017/S0143385700003588.  Google Scholar

[12]

J. Moser, Selected Chapters in the Calculus of Variations, Lectures in Mathematics. ETH Zürich, Birkhäuser, 2003. doi: 10.1007/978-3-0348-8057-2.  Google Scholar

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