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Connecting orbits of Lagrangian systems in a nonstationary force field. (English) Zbl 1353.37126

Summary: We study connecting orbits of a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential \(U(q,t) = f(t)V(q)\). It is assumed that the factor \(f(t)\) tends to \(\infty\) as \(t\to\pm\infty\) and vanishes at a unique point \(t_0\in\mathbb{R}\). Let \(X_+\), \(X_-\) denote the sets of isolated critical points of \(V (x)\) at which \(U(x,t)\) as a function of \(x\) distinguishes its maximum for any fixed \(t > t_{0}\) and \(t < t_{0}\), respectively. Under nondegeneracy conditions on points of \(X_{\pm}\) we prove the existence of infinitely many doubly asymptotic trajectories connecting \(X_-\) and \(X_+\).

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
70H03 Lagrange’s equations
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[1] Bertotti, M.L. and Bolotin, S.V., Doubly Asymptotic Trajectories of Lagrangian Systems in Homogeneous Force Fields, Ann. Mat. Pura Appl. (4), 1998, vol. 174, 253-275. · Zbl 0971.70021 · doi:10.1007/BF01759374
[2] Bessi, U., An Approach to Arnold’s Diffusion through the Calculus of Variations, Nonlinear Anal., 1996, vol. 26, no. 6, pp. 1115-1135. · Zbl 0867.70013 · doi:10.1016/0362-546X(94)00270-R
[3] Bolotin, S.V. and Kozlov, V.V., Asymptotic Solutions of the Equations of Dynamics, Mosc. Univ. Mech. Bull., 1980, vol. 35, nos. 3-4, pp. 82-88; see also: Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1980, no. 4, pp. 84-89, 102. · Zbl 0457.70022
[4] Borisov, A.V., Kozlov, V.V., and Mamaev, I.S., Asymptotic Stability and Associated Problems of Dynamics of Falling Rigid Body, Regul. Chaotic Dyn., 2007, vol. 12, no. 5, pp. 531-565. · Zbl 1229.37107 · doi:10.1134/S1560354707050061
[5] Coti Zelati, V. and Rabinowitz, P.H., Heteroclinic Solutions between Stationary Points at Different Energy Levels, Topol. Methods Nonlinear Anal., 2001, vol. 17, no. 1, pp. 1-21. · Zbl 0984.37073
[6] Giannoni, F., On the Existence of Homoclinic Orbits on Riemannian Manifolds, Ergodic Theory Dynam. Systems, 1994, vol. 14, no. 1, pp. 103-127. · Zbl 0796.58026 · doi:10.1017/S0143385700007744
[7] Giannoni, F. and Rabinowitz, P.H., On the Multiplicity of Homoclinic Orbits on Riemannian Manifolds for a Class of Second Order Hamiltonian Systems, NoDEA Nonlinear Differential Equations Appl., 1994, vol. 1, no. 1, pp. 1-46. · Zbl 0823.34050 · doi:10.1007/BF01194038
[8] Izydorek, M. and Janczewska, J., Heteroclinic Solutions for a Class of the Second Order Hamiltonian Systems, J. Differential Equations, 2007, vol. 238, no. 2, pp. 381-393. · Zbl 1117.37033 · doi:10.1016/j.jde.2007.03.013
[9] Kozlov, V.V., On Falling of a Heavy Rigid Body in an Ideal Fluid, Izv. Akad. Nauk SSSR Mekh. Tverd. Tela, 1989, no. 5, pp. 10-17 (Russian).
[10] Kozlov, V.V., On the Stability of Equilibrium Positions in Non-Stationary Force Fields, J. Appl. Math. Mech., 1991, vol. 55, no. 1, pp. 14-19; see also: Prikl. Mat. Mekh., 1991, vol. 55, no. 1, pp. 12-19. · Zbl 0747.70017 · doi:10.1016/0021-8928(91)90054-X
[11] Palais, R.S., Morse Theory on Hilbert Manifolds, Topology, 1963, vol. 2, 299-340. · Zbl 0122.10702 · doi:10.1016/0040-9383(63)90013-2
[12] Rabinowitz, P.H., Periodic and Heteroclinic Orbits for a Periodic Hamiltonian System, Ann. Inst.H. Poincaré Anal. Non Lineaire, 1989, vol. 6, no. 5, pp. 331-346. · Zbl 0701.58023
[13] Serre, J.-P., Homologie singulière des espaces fibrés. Applications, Ann. of Math. (2), 1951, vol. 54, 425-505. · Zbl 0045.26003 · doi:10.2307/1969485
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