Symmetry breaking in Hamiltonian systems.(English)Zbl 0606.58043

This paper is devoted to the question whether there exist T-periodic solutions of the Hamiltonian system $\dot p=-\frac{\partial H}{\partial q}+\epsilon h_ 1(t)\quad \dot q=\frac{\partial H}{\partial p}+\epsilon h_ 2(t),$ where $$\epsilon >0$$ is small, $$h_ 1$$ and $$h_ 2$$ are T- periodic functions and the unperturbed system $$(H_ 0)$$ is autonomous. The authors give several theorems concerning the question, provided $$(H_ 0)$$ fulfills certain additional assumptions (i.e. convexity, growth conditions, integrability conditions etc.). The method they use to prove these theorems depends on an ”abstract” perturbation result for critical points. This result is proved in the first two sections of the paper.
Reviewer: N.Jacob

MSC:

 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text:

References:

 [1] Albizzati, A, Selection de phase par un terme d’excitation pour LES solutions periodiques de certains équations différentielles, C. R. acad. sci. Paris, 296, 259-262, (1983) · Zbl 0524.34048 [2] Bott, R, Lectures on Morse theory, old and new, Bull. amer. math. soc., 7, 331-358, (1982) · Zbl 0505.58001 [3] Chillingworth, D, Bifurcation from an orbit of symmetry, () · Zbl 0555.58010 [4] Clarke, F, Periodic solutions to Hamiltonian inclusions, J. differential equations, 40, 1-6, (1981) · Zbl 0461.34030 [5] Clarke, F; Ekeland, I, Hamiltonian trajectories having prescribed minimal period, Comm. pure appl. math., 33, 103-116, (1980) · Zbl 0403.70016 [6] Dancer, E.N, The G-invariant implicit function theorem in infinite dimensions, (), 13-30 · Zbl 0512.58011 [7] Ekeland, I, Oscillations de systèmes hamiltoniens non linéaires, III, Bull soc. math. France, 109, 297-330, (1981) · Zbl 0514.70025 [8] Ekeland, I, A perturbation theory near convex Hamiltonian systems, J. differential equations, 50, 407-440, (1983) · Zbl 0476.34035 [9] Ekeland, I, Une théorie de Morse pour LES systèmes hamiltonienes convexes, Ann. inst. H. Poincaré, 1, 143-197, (1984) [10] {\scI. Ekeland and R. Temam}, “Analyse convexe et problèmes variationelles,” Dunod/Gauthier-Villars, Paris. [11] Hardy, G.H; Littlewood, J.E; Polya, G, Inequalities, (1952), Cambridge Univ. Press Cambridge · Zbl 0047.05302 [12] Marino, A; Prodi, G, Metodi perturbativi nella teoria di Morse, Boll. un. mat. ital. (4), 11, 1-32, (1975), Suppl. Fasc. 3 · Zbl 0311.58006 [13] Moser, J, Lectures on Hamiltonian systems, () · Zbl 0172.11401 [14] Palais, R, Lusternik-schnirelman theory on Banach manifolds, Topology, 5, 115-132, (1969) · Zbl 0143.35203 [15] Reeken, M, Stability of critical points under small perturbations. II. analytic theory, Manuscripta math., 8, 69-92, (1973) · Zbl 0248.58004 [16] Spanier, E, Algebraic topology, (1966), McGraw-Hill New York · Zbl 0145.43303 [17] Weinstein, A, Bifurcations and Hamilton’s principle, Math. Z., 159, 235-248, (1978) · Zbl 0366.58003 [18] Vanderbauwhede, A, Local bifurcation and symmetry, () · Zbl 0539.58022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.