# American Institute of Mathematical Sciences

July  2011, 29(3): 1097-1111. doi: 10.3934/dcds.2011.29.1097

## Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds

 1 Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China, China 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona

Received  March 2009 Revised  September 2010 Published  November 2010

Let $\mathcal M$ be a smooth Riemannian manifold with the metric $(g_{ij})$ of dimension $n$, and let $H= 1/2 g^{ij}(q)p_ip_j+V(t,q)$ be a smooth Hamiltonian on $\mathcal M$, where $(g^{ij})$ is the inverse matrix of $(g_{ij})$. Under suitable assumptions we prove the existence of heteroclinic orbits of the induced Hamiltonian systems.
Citation: Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097
##### References:
 [1] V. I. Arnold, "Mathematial Methods of Classical Mechanics," 2nd edition, Springer, New York, 1989. [2] A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology and Classification," Chapman & Hall/CRC, Boca Raton, FL, 2004. [3] K. Burns and M. Gidea, "Differential Geometry and Topology: With a View to Dynamical Systems," Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2005. [4] E. Canalias and J. J. Masdemont, Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems, Discrete Contin. Dyn. Syst., 14 (2006), 261-279. [5] P. C. Carriã and O. H. Miyagaki, Existence of homoclinic solutions for a class of time dependent Hamiltonian systems, J. Math. Anal. Appl., 230 (1999), 157-172. doi: 10.1006/jmaa.1998.6184. [6] Ch. N. Chen and S. Y. Tzeng, Periodic solutions and their connecting orbits of Hamiltonian systems, J. Diff. Eqns., 177 (2001), 121-145. doi: 10.1006/jdeq.2000.3996. [7] C. Chen, F. Liu and X. Zhang, Orthogonal separable Hamitonian systems on $T^2$, Science in China Ser. A, 50 (2007), 1725-1737. doi: 10.1007/s11425-007-0156-7. [8] M. do Carmo, "Riemannian Geometry," Birkhaser, Boston, 1992. [9] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer, Berlin, 1983. [10] M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Diff. Eqns., 219 (2005), 375-389. doi: 10.1016/j.jde.2005.06.029. [11] M. Izydorek and J. Janczewska, Heteroclinic solutions for a class of the second order Hamiltonian systems, J. Diff. Eqns., 238 (2007), 381-393. doi: 10.1016/j.jde.2007.03.013. [12] J. Milnor, "Morse Theory," Princenton University Press, Princenton, 1963. [13] J. Moser, "Selected Chapters in the Calculus of Variations," Birkhäuser, Basel, 2003. [14] P. H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. Inst. H. Poincaré, 6 (1989), 311-346. [15] P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38. [16] P. H. Rabinowitz, Connecting orbits for a reversible Hamiltonian systems, Ergodic Theory Dynam. Systems, 20 (2000), 1767-1784. doi: 10.1017/S0143385700000985. [17] P. H. Rabinowitz, Variational methods for Hamiltonian systems, in "Handbood of Dynamical Sysstems," Vol. 1A, Elsevier, Amsterdam, (2002), 1091-1127. [18] P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 472-499. doi: 10.1007/BF02571356. [19] W. Rudin, "Real and Complex Analysis," 3rd edition, McGraw-Hill Book Co., New York, 1987. [20] A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41. doi: 10.1006/jfan.2001.3798.

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##### References:
 [1] V. I. Arnold, "Mathematial Methods of Classical Mechanics," 2nd edition, Springer, New York, 1989. [2] A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology and Classification," Chapman & Hall/CRC, Boca Raton, FL, 2004. [3] K. Burns and M. Gidea, "Differential Geometry and Topology: With a View to Dynamical Systems," Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2005. [4] E. Canalias and J. J. Masdemont, Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems, Discrete Contin. Dyn. Syst., 14 (2006), 261-279. [5] P. C. Carriã and O. H. Miyagaki, Existence of homoclinic solutions for a class of time dependent Hamiltonian systems, J. Math. Anal. Appl., 230 (1999), 157-172. doi: 10.1006/jmaa.1998.6184. [6] Ch. N. Chen and S. Y. Tzeng, Periodic solutions and their connecting orbits of Hamiltonian systems, J. Diff. Eqns., 177 (2001), 121-145. doi: 10.1006/jdeq.2000.3996. [7] C. Chen, F. Liu and X. Zhang, Orthogonal separable Hamitonian systems on $T^2$, Science in China Ser. A, 50 (2007), 1725-1737. doi: 10.1007/s11425-007-0156-7. [8] M. do Carmo, "Riemannian Geometry," Birkhaser, Boston, 1992. [9] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer, Berlin, 1983. [10] M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Diff. Eqns., 219 (2005), 375-389. doi: 10.1016/j.jde.2005.06.029. [11] M. Izydorek and J. Janczewska, Heteroclinic solutions for a class of the second order Hamiltonian systems, J. Diff. Eqns., 238 (2007), 381-393. doi: 10.1016/j.jde.2007.03.013. [12] J. Milnor, "Morse Theory," Princenton University Press, Princenton, 1963. [13] J. Moser, "Selected Chapters in the Calculus of Variations," Birkhäuser, Basel, 2003. [14] P. H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. Inst. H. Poincaré, 6 (1989), 311-346. [15] P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38. [16] P. H. Rabinowitz, Connecting orbits for a reversible Hamiltonian systems, Ergodic Theory Dynam. Systems, 20 (2000), 1767-1784. doi: 10.1017/S0143385700000985. [17] P. H. Rabinowitz, Variational methods for Hamiltonian systems, in "Handbood of Dynamical Sysstems," Vol. 1A, Elsevier, Amsterdam, (2002), 1091-1127. [18] P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 472-499. doi: 10.1007/BF02571356. [19] W. Rudin, "Real and Complex Analysis," 3rd edition, McGraw-Hill Book Co., New York, 1987. [20] A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41. doi: 10.1006/jfan.2001.3798.
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