October  2013, 33(10): 4769-4793. doi: 10.3934/dcds.2013.33.4769

Ray and heteroclinic solutions of Hamiltonian systems with 2 degrees of freedom

1. 

School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, United States

Received  September 2012 Revised  January 2013 Published  April 2013

In this paper, we study a class of Hamiltonian system with 2 degrees of freedom. We show that at any energy level above a certain critical value of each system, there are ray and heteroclinic solutions between any two periodic neighboring minimal solutions with any prescribed non-trivial homotopy class. Our proof is based on an elementary variational method.
Citation: Guowei Yu. Ray and heteroclinic solutions of Hamiltonian systems with 2 degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4769-4793. doi: 10.3934/dcds.2013.33.4769
References:
[1]

V. I. Arnol'd, "Mathematical Methods of Classical Mechanics,", Second edition. Graduate Texts in Mathematics, (1989). Google Scholar

[2]

V. Bangert, Mather sets for twist maps and geodesics on tori,, in, 1 (1988), 1. Google Scholar

[3]

V. Bangert, Minimal geodesics,, Ergodic Theory Dynam. Systems, 10 (1990), 263. doi: 10.1017/S014338570000554X. Google Scholar

[4]

V. Bangert, Geodesic rays, Busemann functions and monotone twist maps,, Calc. Var. Partial Differential Equations, 2 (1994), 49. doi: 10.1007/BF01234315. Google Scholar

[5]

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

[6]

S. Bolotin, Homoclinic orbits in invariant tori of Hamiltonian systems,, Dynamical systems in classical mechanics, 168 (1995), 21. Google Scholar

[7]

S. Bolotin, Symbolic dynamics near minimal hyperbolic invariant tori of Lagrangian systems,, Nonlinearity, 14 (2001), 1123. doi: 10.1088/0951-7715/14/5/312. Google Scholar

[8]

S. V. Bolotin and P. H. Rabinowitz, Some geometrical conditions for the existence of chaotic geodesics on a torus,, Ergodic Theory Dynam. Systems, 22 (2002), 1407. doi: 10.1017/S0143385702001037. Google Scholar

[9]

G. Contreras and R. Iturriaga, "Global Minimizers of Autonomous Lagrangians,", Available from: , (). Google Scholar

[10]

G, Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values,, Geom. Funct. Anal., 8 (1998), 788. doi: 10.1007/s000390050074. Google Scholar

[11]

G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, The Palais-Smale condition and Mañé's critical values,, Ann. Henri Poincare, 1 (2000), 655. doi: 10.1007/PL00001011. Google Scholar

[12]

G. Contreras, L. Macarini and G. P. Paternain, Periodic orbits for exact magnetic flows on surfaces,, Int. Math. Res. Not., (2004), 361. doi: 10.1155/S1073792804205050. Google Scholar

[13]

G. Contreras and G. P. Paternain, Connecting orbits between static classes for generic Lagrangian systems,, Topology, 41 (2002), 645. doi: 10.1016/S0040-9383(00)00042-2. Google Scholar

[14]

C. Cui, C. Cheng and W. Cheng, Existence of infinitely many homoclinic orbits to Aubry sets for positive definite Lagrangian systems,, J. Differential Equations, 214 (2005), 176. doi: 10.1016/j.jde.2004.08.008. Google Scholar

[15]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", book to appear, (). Google Scholar

[16]

G. A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients,, Ann. of Math., 33 (1932), 719. doi: 10.2307/1968215. Google Scholar

[17]

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

[18]

J. N. Mather, Variational construction of connecting orbits,, Ann. Inst. Fourier, 43 (1993), 1349. doi: 10.5802/aif.1377. Google Scholar

[19]

M, Morse, A fundamental class of geodesics on any closed surface of genus greater than one,, Trans. Amer. Math. Soc., 26 (1924), 25. doi: 10.1090/S0002-9947-1924-1501263-9. Google Scholar

[20]

P. H. Rabinowitz, Heteroclinics for a Hamiltonian system of double pendulum type,, Topol. Methods Nonlinear Anal., 9 (1997), 41. Google Scholar

[21]

P. H. Rabinowitz, A note on heteroclinics for a Hamiltonian system of double pendulum type,, in, 35 (1999), 571. Google Scholar

[22]

P. H. Rabinowitz, Homoclinics and heteroclinics for a Lagrangian system,, Papers in memory of Ennio De Giorgi (Italian). Ricerche Mat. 48 (1999), 48 (1999), 77. Google Scholar

[23]

P. H. Rabinowitz, Solutions of a Lagrangian system on $\mathbbT^2$,, Proc. Natl. Acad. Sci. USA, 96 (1999), 6037. Google Scholar

[24]

P. H. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 673. doi: 10.1016/j.anihpc.2003.10.002. Google Scholar

[25]

P. H. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert. II,, Adv. Nonlinear Stud., 4 (2004), 377. Google Scholar

show all references

References:
[1]

V. I. Arnol'd, "Mathematical Methods of Classical Mechanics,", Second edition. Graduate Texts in Mathematics, (1989). Google Scholar

[2]

V. Bangert, Mather sets for twist maps and geodesics on tori,, in, 1 (1988), 1. Google Scholar

[3]

V. Bangert, Minimal geodesics,, Ergodic Theory Dynam. Systems, 10 (1990), 263. doi: 10.1017/S014338570000554X. Google Scholar

[4]

V. Bangert, Geodesic rays, Busemann functions and monotone twist maps,, Calc. Var. Partial Differential Equations, 2 (1994), 49. doi: 10.1007/BF01234315. Google Scholar

[5]

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

[6]

S. Bolotin, Homoclinic orbits in invariant tori of Hamiltonian systems,, Dynamical systems in classical mechanics, 168 (1995), 21. Google Scholar

[7]

S. Bolotin, Symbolic dynamics near minimal hyperbolic invariant tori of Lagrangian systems,, Nonlinearity, 14 (2001), 1123. doi: 10.1088/0951-7715/14/5/312. Google Scholar

[8]

S. V. Bolotin and P. H. Rabinowitz, Some geometrical conditions for the existence of chaotic geodesics on a torus,, Ergodic Theory Dynam. Systems, 22 (2002), 1407. doi: 10.1017/S0143385702001037. Google Scholar

[9]

G. Contreras and R. Iturriaga, "Global Minimizers of Autonomous Lagrangians,", Available from: , (). Google Scholar

[10]

G, Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values,, Geom. Funct. Anal., 8 (1998), 788. doi: 10.1007/s000390050074. Google Scholar

[11]

G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, The Palais-Smale condition and Mañé's critical values,, Ann. Henri Poincare, 1 (2000), 655. doi: 10.1007/PL00001011. Google Scholar

[12]

G. Contreras, L. Macarini and G. P. Paternain, Periodic orbits for exact magnetic flows on surfaces,, Int. Math. Res. Not., (2004), 361. doi: 10.1155/S1073792804205050. Google Scholar

[13]

G. Contreras and G. P. Paternain, Connecting orbits between static classes for generic Lagrangian systems,, Topology, 41 (2002), 645. doi: 10.1016/S0040-9383(00)00042-2. Google Scholar

[14]

C. Cui, C. Cheng and W. Cheng, Existence of infinitely many homoclinic orbits to Aubry sets for positive definite Lagrangian systems,, J. Differential Equations, 214 (2005), 176. doi: 10.1016/j.jde.2004.08.008. Google Scholar

[15]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", book to appear, (). Google Scholar

[16]

G. A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients,, Ann. of Math., 33 (1932), 719. doi: 10.2307/1968215. Google Scholar

[17]

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

[18]

J. N. Mather, Variational construction of connecting orbits,, Ann. Inst. Fourier, 43 (1993), 1349. doi: 10.5802/aif.1377. Google Scholar

[19]

M, Morse, A fundamental class of geodesics on any closed surface of genus greater than one,, Trans. Amer. Math. Soc., 26 (1924), 25. doi: 10.1090/S0002-9947-1924-1501263-9. Google Scholar

[20]

P. H. Rabinowitz, Heteroclinics for a Hamiltonian system of double pendulum type,, Topol. Methods Nonlinear Anal., 9 (1997), 41. Google Scholar

[21]

P. H. Rabinowitz, A note on heteroclinics for a Hamiltonian system of double pendulum type,, in, 35 (1999), 571. Google Scholar

[22]

P. H. Rabinowitz, Homoclinics and heteroclinics for a Lagrangian system,, Papers in memory of Ennio De Giorgi (Italian). Ricerche Mat. 48 (1999), 48 (1999), 77. Google Scholar

[23]

P. H. Rabinowitz, Solutions of a Lagrangian system on $\mathbbT^2$,, Proc. Natl. Acad. Sci. USA, 96 (1999), 6037. Google Scholar

[24]

P. H. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 673. doi: 10.1016/j.anihpc.2003.10.002. Google Scholar

[25]

P. H. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert. II,, Adv. Nonlinear Stud., 4 (2004), 377. Google Scholar

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