eBook ISBN:  9781470403737 
Product Code:  MEMO/163/775.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 
eBook ISBN:  9781470403737 
Product Code:  MEMO/163/775.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 163; 2003; 145 ppMSC: Primary 70; 37; 34
In this text we take up the problem of the splitting of invariant manifolds in multidimensional Hamiltonian systems, stressing the canonical features of the problem. We first conduct a geometric study, which for a large part is not restricted to the perturbative situation of nearintegrable systems. This point of view allows us to clarify some previously obscure points, in particular the symmetry and variance properties of the splitting matrix (indeed its very definition(s)) and more generally the connection with symplectic geometry. Using symplectic normal forms, we then derive local exponential upper bounds for the splitting matrix in the perturbative analytic case, under fairly general circumstances covering in particular resonances of any multiplicity. The next technical input is the introduction of a canonically invariant scheme for the computation of the splitting matrix. It is based on the familiar HamiltonJacobi picture and thus again is symplectically invariant from the outset. It is applied here to a standard Hamiltonian exhibiting many of the important features of the problem and allows us to explore in a unified way the question of finding lower bounds for the splitting matrix, in particular that of justifying a first order computation (the socalled PoincaréMelnikov approximation). Although we do not specifically address the issue in this paper we mention that the problem of the splitting of the invariant manifold is wellknown to be connected with the existence of a global instability in these multidimensional Hamiltonian systems and we hope the present study will ultimately help shed light on this important connection first noted and explored by V. I. Arnold.
ReadershipGraduate students and research mathematicians interested in geometry, topology, and analysis.

Table of Contents

Chapters

0. Introduction and some salient features of the model Hamiltonian

1. Symplectic geometry and the splitting of invariant manifolds

2. Estimating the splitting matrix using normal forms

3. The Hamilton–Jacobi method for a simple resonance


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
In this text we take up the problem of the splitting of invariant manifolds in multidimensional Hamiltonian systems, stressing the canonical features of the problem. We first conduct a geometric study, which for a large part is not restricted to the perturbative situation of nearintegrable systems. This point of view allows us to clarify some previously obscure points, in particular the symmetry and variance properties of the splitting matrix (indeed its very definition(s)) and more generally the connection with symplectic geometry. Using symplectic normal forms, we then derive local exponential upper bounds for the splitting matrix in the perturbative analytic case, under fairly general circumstances covering in particular resonances of any multiplicity. The next technical input is the introduction of a canonically invariant scheme for the computation of the splitting matrix. It is based on the familiar HamiltonJacobi picture and thus again is symplectically invariant from the outset. It is applied here to a standard Hamiltonian exhibiting many of the important features of the problem and allows us to explore in a unified way the question of finding lower bounds for the splitting matrix, in particular that of justifying a first order computation (the socalled PoincaréMelnikov approximation). Although we do not specifically address the issue in this paper we mention that the problem of the splitting of the invariant manifold is wellknown to be connected with the existence of a global instability in these multidimensional Hamiltonian systems and we hope the present study will ultimately help shed light on this important connection first noted and explored by V. I. Arnold.
Graduate students and research mathematicians interested in geometry, topology, and analysis.

Chapters

0. Introduction and some salient features of the model Hamiltonian

1. Symplectic geometry and the splitting of invariant manifolds

2. Estimating the splitting matrix using normal forms

3. The Hamilton–Jacobi method for a simple resonance