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Full Description
Nonlinear functional analysis has proven to be a powerful alternative to classical perturbation methods in the study of periodic motions of regular Hamiltonian systems. The authors of this monograph present a summary and synthesis of recent research demonstrating that variational methods can be used to successfully handle systems with singular potential, the Lagrangian systems. The classical cases of the Kepler problem and the N-body problem are used as specific examples. Critical point theory is used to obtain existence results, qualitative in nature, which hold true for broad classes of potentials. These results give a functional frame for systems with singular potential. The authors have provided some valuable methods and tools to researchers working on this constantly evolving topic. At the same time, they present the new approach and results that they have shared over recent years with their colleagues and graduate students.
Contents
I. Preliminaries. 1 Lagrangian systems with smooth potentials. 2 Models involving singular Lagrangians. 2.a Kepler's problem. 2.b A class of model potentials. 2.c The N-body problem. 2.d Other problems arising in Celestial Mechanics. 2.e Electricl forces. 3 Critical point theory. II Singular Potentials. 4 The functional setting. 4.a Prescribed period. 4.b Fixed energy. 5 The Strong Force assumption. 6 Collision solutions. III The Strongly Attractive Case. 7 The abstract setting. 7.a The (PS) condition. 7. b The topology of fe. 7.c A critical point theorem. 7.d Another critical point theorem. 8 Existence of periodic solutions. 8. a Even and planar potentials. 8. b The general case. 9 Repulsive potentials. IV The Weakly Attractive Case. 10 Weak solutions. 11 Existence of weak solutions. 12 Regularity of weak solutions. 13 Local assumptions 14 Global assumptions. V Orbits with Prescribed Energy. 15 Strongly attractive potentials. 16 Weakly attractive potentials. 16.a A modified variational principle. 16.b Existence of closed orbits. 17 Symmetric potentials. VI The N-Body Problem. 18 The N-body equation. 19 Even potentials. 20 The general case. 21 Fixed energy. VII Perturbation Results. 22 A perturbation result in critical point theory. 23 T-periodic solutions. 24 First order systems. 25 Solutions of prescribed energy. 26 Restricted N-Body problems.
