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Some results on connecting orbits for a class of Hamiltonian systems. (English) Zbl 0707.58022
The existence of various kinds of connecting orbits is established for the Hamiltonian system $$(HS)\quad q''+V'(q)=0$$ as well as its time periodic analogue. For the autonomous case, the main assumption is that V has a global maximum, e.g. at $$x=0$$. Variational methods then establish the existence of various kinds of orbits terminating at $$x=0$$. For the time dependent case it is assumed that V has a local but not global maximum at $$x=0$$ and it is proved that (HS) has a homoclinic orbit emanating from and terminating at 0.
Reviewer: P.H.Rabinowitz

##### MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 58E30 Variational principles in infinite-dimensional spaces
##### Keywords:
connecting orbits; Hamiltonian system
Full Text:
##### References:
 [1] Rabinowitz, P.H.: Periodic and heteroclinic orbits for a periodic Hamiltonian system. Analyse Nonlineaire6, 331–346 (1989) · Zbl 0701.58023 [2] Kozlov, V.V.: Calculus of variations in the large and classical mechanics. Russ. Math. Surv.40, 37–71 (1985) · Zbl 0579.70020 [3] Coti-Zelati, V., Ekeland, I., Sere, E.: A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann.288, 133–160 (1990) · Zbl 0731.34050 [4] Hofer, H., Wysocki, K.: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. (preprint) · Zbl 0702.34039 [5] Rabinowitz, P.H.: Homoclinic orbits for a class of Hamiltonian systems. Proc. Royal Soc. Edinburgh114A, 33–38 (1990) · Zbl 0705.34054 [6] Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian systems. Berlin Heidelberg New York: Springer 1989 · Zbl 0676.58017 [7] Nehari, Z.: On a class of nonlinear second order differential equations. Trans. Am. Math. Soc.95, 101–123 (1960) · Zbl 0097.29501 [8] Nehari, Z.: Characteristic values associated with a class of nonlinear second-order differential equations. Acta Math.105, 141–175 (1961) · Zbl 0099.29104 [9] Coffman, C.: A minimum-maximum principle for a class of nonlinear integral equations. J. Anal. Math.22, 391–419 (1969) · Zbl 0179.15601 [10] Hempel, J.A.: Superlinear boundary value problems and nonuniqueness. University of New England, Armidale thesis 1970
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