Perturbation Theory: Mathematics, Methods and Applications

Series Preface
Volume Preface
Contents
About the Editor-in-Chief
About the Volume Editor
Contributors
Perturbation Theory
Glossary
Definition of the Subject
Introduction
Poincaré´s Theorem and Quanta
Mathematics and Physics. Renormalization
Need of Convergence Proofs
Multiscale Analysis
A Paradigmatic Example of PT Problem
Lindstedt Series
Convergence. Scales. Multiscale Analysis
Non Convergent Cases
Conclusion and Outlook
Future Directions
Bibliography
Hamiltonian Perturbation Theory (and Transition to Chaos)
Glossary
Definition of the Subject
Introduction
The Perturbation Problem
Questions of Persistence
General Dynamics
Chaos
One Degree of Freedom
Hamiltonian Perturbations
Dissipative Perturbations
Reversible Perturbations
Perturbations of Periodic Orbits
Conservative Perturbations
Dissipative Perturbations
Invariant Curves of Planar Diffeomorphisms
Circle Maps
Area-Preserving Maps
Linearization of Complex Maps
Complex Linearization
Cremer´s Example in Herman´s Version
KAM Theory: An Overview
Classical KAM Theory
Dissipative KAM Theory
Lower Dimensional Tori
Global KAM Theory
Splitting of Separatrices
Periodic Orbits
(n-1)-Tori
Transition to Chaos and Turbulence
Quasi-periodic Bifurcations
Hamiltonian Cases
Dissipative Cases
A Scenario for the Onset of Turbulence
Future Directions
Bibliography
Perturbation Theory in Quantum Mechanics
Glossary
Definition of the Subject
Introduction
Presentation of the Problem and an Example
Perturbation of Point Spectra: Nondegenerate Case
Corrections to the Energy and the Eigenvectors
Wigner´s Theorem
The Feynman-Hellmann Theorem
Perturbation of Point Spectra: Degenerate Case
Corrections to the Energy and the Eigenvectors
Bloch´s Method
The Quasi-degenerate Case
The Brillouin-Wigner Method
Symmetry and Degeneracy
Symmetry and Perturbation Theory
Level Crossing
Problems with the Perturbation Series
Perturbation of the Continuous Spectrum
Scattering Solutions and Scattering Amplitude
The Born Series and Its Convergence
Time-Dependent Perturbations
Future Directions
Bibliography
Books and Reviews
Normal Forms in Perturbation Theory
Glossary
Definition of the Subject
Introduction
Motivation
Reduction of Toroidal Symmetry
A Global Perturbation Theory
The Normal Form Procedure
Background, Linearization
Preliminaries from Differential Geometry
`Simple´ in Terms of anAdjoint Action
Torus Symmetry
On the Choices of the Complementary Space and of the Normalizing Transformation
Preservation of Structure
The Lie-Algebra Proof
The Volume Preserving and Symplectic Case
External Parameters
The Reversible Case
The Hamiltonian Case
Semi-local Normalization
A Diffeomorphism Near a Fixed Point
Near a Periodic Solution
Near a Quasi-periodic Torus
Non-formal Aspects
Normal Form Symmetry and Genericity
On Convergence
Applications
`Cantorized´ Singularity Theory
On the Averaging Theorem
Future Directions
Bibliography
Primary Literature
Books and Reviews
Convergence of Perturbative Expansions
Glossary
Definition of the Subject
Introduction
Poincaré-Dulac Normal Forms
Convergence and Convergence Problems
Lie Algebra Arguments
NFIM and Sets of Analyticity
Hamiltonian Systems
Future Directions
Bibliography
Diagrammatic Methods in Classical Perturbation Theory
Glossary
Definition of the Subject
Introduction
Examples
A Class of Quasi-integrable Hamiltonian Systems
A Simplified Model with No Small Divisors
The Standard Map
Trees and Graphical Representation
Trees
Labels and Diagrammatic Rule
Small Divisors
Multiscale Analysis
Resummation
Generalizations
Lower-Dimensional Tori
Other Ordinary Differential Equations
Bryuno Vectors
Partial Differential Equations
Conclusions and Future Directions
Bibliography
Primary Literature
Books and Reviews
Perturbation Theory and the Method of Detuning
Glossary
Introduction
Classical Detuning
Normalization
Quasi-resonant Normalization
Detuned Resonant 2-DOF Systems
Variables Adapted to the k: Resonance
Classical Examples
The Symmetric 1:1 Resonance
The 2:1 Resonance
Future Directions
Bibliography
Primary Literature
Books and Reviews
Computational Methods in Perturbation Theory
Glossary
Definition of the Subject
Introduction
The Solar System
Dynamics Near an Equilibrium Point of a Hamiltonian System
Time-Dependent Perturbations
Quasi-Periodic Motions and KAM Theory
The Parametrization Method
Other Situations
Future Directions
Acknowledgments
Bibliography
Perturbation Analysis of Parametric Resonance
Glossary
Definition of the Subject
Introduction
Perturbation Techniques
Poincaré-Lindstedt Series
Averaging
Resonance
Normalization of Time-Dependent Vectorfields
Remarks on Limit Sets
Parametric Excitation of Linear Systems
Elementary Theory
Higher Order Approximation and an Unexpected Timescale
The Mathieu Equation with Viscous Damping
Coexistence
More General Classical Results
Quasi-Periodic Excitation
Parametrically Forced Oscillators in Sum Resonance
Nonlinear Parametric Excitation
The Conservative Case, κ = 0
Adding Dissipation, κ > 0
Coexistence Under Nonlinear Perturbation
Other Nonlinearities
Applications
The Parametrically Excited Pendulum
Rotor Dynamics
Instability by Damping
Autoparametric Excitation
Future Directions
Acknowledgments
Bibliography
Primary Literature
Books and Reviews
Symmetry and Perturbation Theory in Non-linear Dynamics
Glossary
Definition of the Subject
Introduction
Symmetry of Dynamical Systems
Perturbation Theory: Normal Forms
Poincaré-Dulac Normal Forms
Lie Transforms
Perturbative Determination of Symmetries
Determining Equations
Recursive Solution of the Determining Equations
Approximate Symmetries
Symmetry Characterization of Normal Forms
Linear Algebra
Normal Forms
The General Case
Symmetries and Transformation to Normal Form
Nonlinear Symmetries (The General Case)
Linear Symmetries
Generalizations
Abelian Lie Algebra
Nilpotent Lie Algebra
General Lie Algebra
Symmetry for Systems in Normal Form
Linearization of a Dynamical System
Further Normalization and Symmetry
Further Normalization and Resonant Further Symmetry
Further Normalization and External Symmetry
Symmetry Reduction of Symmetric Normal Forms
Conclusions
Future Developments
Additional Notes
Bibliography
Perturbation of Systems with Nilpotent Real Part
Glossary
Definition of the Subject
Introduction
Complex and Real Jordan Canonical Forms
Nilpotent Perturbation and Formal Normal Forms of Vector Fields and Maps Near a Fixed Point
Loss of Gevrey Regularity in Siegel Domains in the Presence of Jordan Blocks
First-Order Singular Partial Differential Equations
Normal Forms for Real Commuting Vector Fields with Linear Parts Admitting Nontrivial Jordan Blocks
Analytic Maps near a Fixed Point in the Presence of Jordan Blocks
Weakly Hyperbolic Systems and Nilpotent Perturbations
Bibliography
Perturbation Theory for PDEs
Glossary
Definition of the Subject
Introduction
The Hamiltonian Formalism for PDEs
The Gradient of a Functional
Lagrangian and Hamiltonian Formalism for the Wave Equation
Canonical Coordinates
Basic Elements of Hamiltonian Formalism for PDEs
Normal Form for Finite Dimensional Hamiltonian Systems
Normal Form for Hamiltonian PDEs: General Comments
Normal Form for Resonant Hamiltonian PDEs and Its Consequences
Normal Form for Nonresonant Hamiltonian PDEs
A Statement
Verification of the Property of Localization of Coefficients
Verification of the Nonresonance Property
Non Hamiltonian PDEs
Extensions and Related Results
Future Directions
Bibliography
Kolmogorov-Arnold-Moser (KAM) Theory for Finite and Infinite Dimensional Systems
Glossary
Definition of the Subject
Introduction
Finite Dimensional Context
Infinite Dimensional Context
Finite Dimensional KAM Theory
Kolmogorov Theorem
Step 1: Kolmogorov Transformation
Step 2: Estimates
Step 3: Iteration and Convergence
Arnold´s Scheme
The Differentiable Case: Moser´s Theorem
Lower Dimensional KAM Tori
Other Chapters in Classical KAM Theory
Infinite Dimensional KAM Theory
Future Directions
Appendix A: The Classical Implicit Function Theorem
Appendix B: Complementary Notes
Bibliography
Books and Reviews
Nekhoroshev Theory
Glossary
Definition of the Subject
Introduction
Integrable and Quasi-Integrable Hamiltonian Systems
Averaging Principle
Hamiltonian Perturbation Theory
Exponential Stability of Constant Frequency Systems
The Case of a Single Frequency System
The Case of a Strongly Nonresonant Constant Frequency System
Nekhoroshev Theory (Global Stability)
The Initial Statement
Improved Versions of Nekhoroshev Theorem
KAM Stability, Exponential Stability, Nekhoroshev Stability
Applications
The Case of a Constant Frequency Integrable System
Application of the Global Nekhoroshev Theory
Future Directions
Appendix A
An Example of Divergence Without Small Denominators
Bibliography
Primary Literature
Books and Reviews
Perturbation of Superintegrable Hamiltonian Systems
Glossary
Definition of the Subject
Introduction
Superintegrable Hamiltonian Systems
A Paradigm for Integrability
The Hamiltonian Case
The Symplectic Structure of Noncommutatively Integrable Systems
The Geometric Structure of Integrable Toric Fibrations
Global Questions
Dynamics
Geometric Characterization of Superintegrability
Examples
Perturbations of Superintegrable Systems
Introduction
Semiglobal Approach
Nekhoroshev Theorem for Superintegrable Systems
Motions Along the Symplectic Leaves
KAM Theory
Some Applications
Conclusion
Appendix: Semiglobal Normal Forms
Acknowledgments
Bibliography
Perturbation Theory in Celestial Mechanics
Glossary
Definition of the Subject
Introduction
Classical Perturbation Theory
The Classical Theory
The Precession of the Perihelion of Mercury
Delaunay Action: Angle Variables
The Restricted, Planar, Circular, Three-Body Problem
Expansion of the Perturbing Function
Computation of the Precession of the Perihelion
Resonant Perturbation Theory
The Resonant Theory
Three-Body Resonance
Degenerate Perturbation Theory
The Precession of the Equinoxes
Invariant Tori
Invariant KAM Surfaces
Rotational Tori for the Spin-Orbit Problem
Librational Tori for the Spin-Orbit Problem
Rotational Tori for the Restricted Three-Body Problem
Planetary Problem
Periodic Orbits
Construction of Periodic Orbits
The Libration in Longitude of the Moon
Future Directions
Bibliography
n-Body Problem and Choreographies
Glossary
Definition of the Subject
Introduction
Singular Hamiltonian Systems
Simple Choreographies and Relative Equilibria
Basic Definitions and Notations
Symmetry Groups and Equivariant Orbits
Cyclic and Dihedral Actions
The Variational Approach
The Eight Shaped Three-Body Solution
The Rotating Circle Property (RCP)
More Examples with non Trivial Core
The 3-Body Problem
The Classification of Planar Symmetry Groups for 3-body
Space Three-body Problem
Minimizing Properties of Simple Choreographies
When ω is Close to an Integer
Mountain Pass Solutions for the Choreographical 3-Body Problem
Generalized Orbits and Singularities
Singularities and Collisions
The Theorems of Painlevé and Von Zeipel
Von Zeipel´sTheorem and the Structure of the Collision Set
Asymptotic Estimates at Collisions
One Side Conditions on the Potential and Its Radial Derivative
Isolatedness of Collisions Instants
Conservation Laws
Generalized Sundman-Sperling Estimates
Dissipation and McGehee Coordinates
Blow-ups
Logarithmic Type Potentials
Absence of Collision for Locally Minimal Paths
Quasi-Homogeneous Potentials
Neumann Boundary Conditions and G-equivariant Minimizers
The Standard Variation
Some Properties of Φ
Future Directions
Bibliography
Primary Literature
Books and Reviews
Semiclassical Perturbation Theory
Glossary
Definition of the Subject
Introduction
Notation
The WKB Approximation
Semiclassical Solutions in the Classically Allowed Region
Semiclassical Solutions in the Classically Forbidden Region
Connection Formula
The Complex Method
The Method of Comparison Equations
Bound States for a Single Well Potential
Double Well Model: Estimate of theSplitting and the ``Flea of the Elephant´´
Semiclassical Approximation in Any Dimension
Semiclassical Eigenvalues at the Bottom of a Well
Agmon Metric
Tunneling Between Wells
Propagation of Quantum Observables
Brief Review of -Pseudodifferential Calculus
Egorov Theorem
Future Directions
Bibliography
Perturbation Theory and Molecular Dynamics
Glossary
Definition of the Subject
Introduction
The Framework
The Leading Order Born-Oppenheimer Approximation
Beyond the Leading Order
Future Directions
Bibliography
Primary Literature
Books and Reviews
Quantum Adiabatic Theorem
Glossary
Definition of the Subject and Its Importance
Introduction
Kato´s Quantum Adiabatic Theorem
Berry´s Connection and Parallel Transport
Super-Adiabatic Expansions
Exponentially Small Nonadiabatic Transitions
Generalizations and Further Aspects
Space-Adiabatic Theorems
Adiabatic Theorems Without Gap Condition
Adiabatic Theorems for Resonances
Adiabatic Theorems for Open Quantum Systems
Adiabatic Theorems for Many-Body Quantum Systems
Adiabatic Theorems for Nonlinear Dynamics
Future Directions
Bibliography
Books
Quantum Bifurcations
Glossary
Definition of the Subject
Introduction
Simplest Effective Hamiltonians
Simplest Hamiltonians for Two Degree-of-Freedom Systems
Bifurcations and Symmetry
Imperfect Bifurcations
Organization of Bifurcations
Bifurcation Diagrams for Two Degree-of-Freedom Integrable Systems
Bifurcations of ``Quantum Bifurcation Diagrams´´
Semi-Quantum Limit and Reorganization of Quantum Bands
Multiple Resonances and Quantum State Density
Physical Applications and Generalizations
Future Directions
Bibliography
Convergent Perturbative Expansion in Condensed Matter and Quantum Field Theory
Glossary
Definition of the Subject and Its Importance
Gaussian Integrals
Grassmann Integrals
Perturbative Expansions
Truncated Expectations
Analyticity
Conclusions
Biblilography
Correlation Corrections as a Perturbation to the Quasi-free Approximation in Many-Body Quantum Systems
Glossary
Definition of the Subject
Introduction
The Framework of Quantum Mechanics
Many-Body Quantum Mechanics
Quantities of Interest
Article Roadmap
Scaling Limits
Bosons: Low-Density Scaling Limit (Gross-Pitaevskii Limit)
Fermions: High-Density Scaling Limit (Mean-Field/Semiclassical Limit)
Second Quantization
Bogoliubov Transformations and Quasi-free States
Quadratic Hamiltonians
Quasi-free Approximations
Bosons: Gross-Pitaevskii Approximation
Fermions: Hartree-Fock Approximation
Correlation Corrections to the Gross-Pitaevskii Approximation
Correlation Corrections to the Hartree-Fock Approximation
Theory of Correlation Corrections for Fermions
Future Directions
Bosons
Fermions
References
Perturbation of Equilibria in the Mathematical Theory of Evolution
Glossary
Definition of the Subject
Introduction
Evolution on a Fitness Landscape
Stability of Equilibria on a Fitness Landscape
Perturbation of Equilibria on a Fitness Landscape
Frequency Dependent Fitness: Game Theory
Equilibria in Evolutionary Game Theory
Perturbations of Equilibria in Evolutionary Game Theory
Spatial Perturbations
Time Scales
Future Directions
Bibliography
Primary Literature
Books and Reviews
Perturbation Theory for Non-smooth Systems
Glossary
Definition of the Subject
Introduction
Preliminaries
Discontinuous Systems
Singular Perturbation Problem
Regularization Process
Vector Fields Near the Boundary
A Construction
Codimension-one M-Singularity in Dimensions Two and Three
Generic Bifurcation
Two-Dimensional Case
Three-Dimensional Case
Singular Perturbation Problem in 2D
Future Directions
Some Problems
Conclusion
Bibliography
Primary Literature
Books and Reviews
Exact and Perturbation Methods in the Dynamics of Legged Locomotion
Glossary
Definition of the Subject
Introduction
Poincaré Maps for Systems with Impacts
The Planar Rimless Wheel (Poincaré Map Is Explicitly Computable)
The Actuated Planar Biped (Poincaré Map Reduces to an Explicitly Computable Map)
Fixed Points of Perturbed Poincaré Maps
Compass-Gait Biped (Poincare Map Is a Perturbation of an Explicitly Computable Map)
Future Directions
Nonperiodic Motions (Existence of Invariant Tori)
Varying Impact Law (Dependence of Stable Manifolds on Parameters)
The Effect of Leg Scuffing on Stability of the Walking Cycle (Grazing Bifurcations)
The Effect of Soft Ground on Stability of the Walking Cycle (Singular Perturbations)
The Effect of Accounting for the Third Dimension (Higher-Dimensional Perturbation Analysis)
Acknowledgments
Bibliography
Perturbation Theory for Water Waves
Glossary
Definition of the Subject
Introduction
KAM Results for Water Waves
The KAM for Gravity Capillary Water Waves
Some Ideas of the Proof
The KAM for Pure Gravity Water Waves in Finite Depth
Ideas of the Proof
Quasi-periodic Traveling Water Waves
Longtime Existence for Periodic Water Waves
Birkhoff Normal Form and Longtime Existence for Gravity Capillary Water Waves
The Dyachenko-Zakharov Conjecture for Pure Gravity Water Waves
Future Developments
Bibliography
Periodic Rogue Waves and Perturbation Theory
Glossary
Definition of the Subject
Introduction
The Finite Gap Method and the Periodic NLS Cauchy Problem of the Rogue Waves, for a Finite Number of Unstable Modes
Periodic Problem for the Focusing Nonlinear Schrödinger Equation
Finite-Gap Approximation
Cauchy Problem for the RWs
The Spectral Data for the Unperturbed Operator
The Spectral Data for the Perturbed Operator
The Leading Order Finite-Gap Solution
The Solution of the Cauchy Problem in Terms of Elementary Functions
Keeping Only Visible Modes
The Case N = 1 and the Fermi-Pasta-Ulam-Tsingou Recurrence of RWs
RW Perturbation Theory
Future Directions
References
Index