Wave breaking and global existence for the periodic rotation-Camassa-Holm system
Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth
April 2017, 37(4): 2227-2242. doi: 10.3934/dcds.2017096
Suspension of the billiard maps in the Lazutkin's coordinate
Jianlu Zhang
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, M5S 2E4, Canada
Received July 2016 Revised September 2016 Published December 2016
Full Text(HTML)
Figure(2)
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
$H(x,p,t)=p^{3/2}+p^{5/2}V(x,p^{1/2},t),\;\;(x,p,t)∈\mathbb{T}×[0,+∞)×\mathbb{T},$
$ V(·,·,·) $
$ 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.
Keywords: Billiard maps, Hamiltonian system, Aubry Mather theory, symplectic suspension.
Mathematics Subject Classification: Primary:37J50;Secondary:70H08.
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
P. Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier, Grenoble, 52 (2002), 1533-1568. doi: 10.5802/aif.1924. Google Scholar
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
C.-Q. Cheng and J. Yan, Arnold diffusion in hamiltonian systems: A priori unstable case, J. Differential Geometry, 82 (2009), 229-277. Google Scholar
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
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
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
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
J. Mather, Glancing billiards, Ergod. Th. Dyn. Sys., 2 (1982), 397-403. doi: 10.1017/S0143385700001681. Google Scholar
J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Mathematische Zeitschrift, 207 (1991), 169-207. doi: 10.1007/BF02571383. Google Scholar
J. Mather, Variational construction of connecting orbits, Annales de l'institut Fourier, 43 (1993), 1349-1386. doi: 10.5802/aif.1377. Google Scholar
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
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
Figure Options
Download as PowerPoint slide
Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018
Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135
Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807
Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683
Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155
Bassam Fayad. Discrete and continuous spectra on laminations over Aubry-Mather sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 823-834. doi: 10.3934/dcds.2008.21.823
Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103
Siniša Slijepčević. The Aubry-Mather theorem for driven generalized elastic chains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983
P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1
Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077
Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
Jianlu Zhang. Coexistence of period 2 and 3 caustics for deformative nearly circular billiard maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6419-6440. doi: 10.3934/dcds.2019278
Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325
Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431
Carles Simó, Dmitry Treschev. Stability islands in the vicinity of separatrices of near-integrable symplectic maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 681-698. doi: 10.3934/dcdsb.2008.10.681
Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
L. Búa, T. Mestdag, M. Salgado. Symmetry reduction, integrability and reconstruction in $k$-symplectic field theory. Journal of Geometric Mechanics, 2015, 7 (4) : 395-429. doi: 10.3934/jgm.2015.7.395
Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684
Katarzyna Grabowska. Lagrangian and Hamiltonian formalism in Field Theory: A simple model. Journal of Geometric Mechanics, 2010, 2 (4) : 375-395. doi: 10.3934/jgm.2010.2.375
HTML views (13)