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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 and Continuous Dynamical Systems, 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.

[2]

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

[3]

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

[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. 

[5]

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

[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. 

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[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.

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.

[2]

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

[3]

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

[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. 

[5]

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

[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. 

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[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.

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