Electronic ISBN:  9781470402471 
Product Code:  MEMO/138/658.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 138; 1999; 112 ppMSC: 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 PalaisSmale 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 4body type problems then generalized to nbody type problems.ReadershipGraduate students and research mathematicians working in applications of Morse theory and the study of dynamical systems.

Table of Contents

Chapters

1. Introduction

2. Breakdown of the PalaisSmale condition

3. Morse lemma near infinity

4. A modified functional for the 4body problem

5. Retraction theorem and related results for the 4body problem

6. Generalization to the $n$body problem


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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 PalaisSmale 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 4body type problems then generalized to nbody 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 PalaisSmale condition

3. Morse lemma near infinity

4. A modified functional for the 4body problem

5. Retraction theorem and related results for the 4body problem

6. Generalization to the $n$body problem