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Study of the Critical Points at Infinity Arising from the Failure of the Palais-Smale Condition for n-Body Type Problems
Hasna Riahi, Ecole Nationale d'Ingénieurs de Tunis, Tunisia

Memoirs of the American Mathematical Society
1999; 112 pp; softcover
Volume: 138
ISBN-10: 0-8218-0873-7
ISBN-13: 978-0-8218-0873-3
List Price: US$50
Individual Members: US$30
Institutional Members: US$40
Order Code: MEMO/138/658
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In this work, the author examines the following: When the Hamiltonian system \(m_i \ddot{q}_i + (\partial V/\partial q_i) (t,q) =0\) with periodicity condition \(q(t+T) = q(t),\; \forall t \in \mathfrak R\) (where \(q_{i} \in \mathfrak R^{\ell}\), \(\ell \ge 3\), \(1 \le i \le n\), \(q = (q_{1},...,q_{n})\) and \(V = \sum V_{ij}(t,q_{i}-q_{j})\) with \(V_{ij}(t,\xi)\) \(T\)-periodic in \(t\) and singular in \(\xi\) at \(\xi = 0\)) is posed as a variational problem, the corresponding functional does not satisfy the Palais-Smale condition and this leads to the notion of critical points at infinity.

This volume is a study of these critical points at infinity and of the topology of their stable and unstable manifolds. The potential considered here satisfies the strong force hypothesis which eliminates collision orbits. The details are given for 4-body type problems then generalized to n-body type problems.


Graduate students and research mathematicians working in applications of Morse theory and the study of dynamical systems.

Table of Contents

  • Introduction
  • Breakdown of the Palais-Smale condition
  • Morse Lemma near infinity
  • A modified functional for the 4-body problem
  • Retraction theorem and related results for the 4-body problem
  • Generalization of the n-body problem
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