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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
Available Formats:
Electronic ISBN: 978-1-4704-0247-1
Product Code: MEMO/138/658.E
112 pp
List Price: $50.00 MAA Member Price:$45.00
AMS Member Price: $30.00 Click above image for expanded view 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 Available Formats:  Electronic ISBN: 978-1-4704-0247-1 Product Code: MEMO/138/658.E 112 pp  List Price:$50.00 MAA Member Price: $45.00 AMS Member Price:$30.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1381999
MSC: Primary 58; Secondary 34; 70;

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.

• Chapters
• 1. Introduction
• 2. Breakdown of the Palais-Smale condition
• 3. Morse lemma near infinity
• 4. A modified functional for the 4-body problem
• 5. Retraction theorem and related results for the 4-body problem
• 6. Generalization to the $n$-body problem
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Volume: 1381999
MSC: Primary 58; Secondary 34; 70;

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.

• 6. Generalization to the $n$-body problem