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gaeta g perturbation theory mathematics methods app 2ed 2022

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Textbook in PDF format About the Editor-in-Chief About the Volume Editor Perturbation Theory Definition of the Subject 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 Hamiltonian Perturbation Theory (and Transition to Chaos) Definition of the Subject 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 Perturbation Theory in Quantum Mechanics Definition of the Subject 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 Books and Reviews Normal Forms in Perturbation Theory Definition of the Subject 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 Primary Literature Books and Reviews Convergence of Perturbative Expansions Definition of the Subject Poincaré-Dulac Normal Forms Convergence and Convergence Problems Lie Algebra Arguments NFIM and Sets of Analyticity Hamiltonian Systems Future Directions Diagrammatic Methods in Classical Perturbation Theory Definition of the Subject 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 Primary Literature Books and Reviews Perturbation Theory and the Method of Detuning 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 Primary Literature Books and Reviews Computational Methods in Perturbation Theory Definition of the Subject 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 Perturbation Analysis of Parametric Resonance Definition of the Subject 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 Primary Literature Books and Reviews Symmetry and Perturbation Theory in Non-linear Dynamics Definition of the Subject 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 Perturbation of Systems with Nilpotent Real Part Definition of the Subject 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 Perturbation Theory for PDEs Definition of the Subject 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 Kolmogorov-Arnold-Moser (KAM) Theory for Finite and Infinite Dimensional Systems Definition of the Subject 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 Books and Reviews Nekhoroshev Theory Definition of the Subject 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 Primary Literature Books and Reviews Perturbation of Superintegrable Hamiltonian Systems Definition of the Subject 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 Semiglobal Approach Nekhoroshev Theorem for Superintegrable Systems Motions Along the Symplectic Leaves KAM Theory Some Applications Appendix: Semiglobal Normal Forms Perturbation Theory in Celestial Mechanics Definition of the Subject 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 n-Body Problem and Choreographies Definition of the Subject 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 Primary Literature Books and Reviews Semiclassical Perturbation Theory Definition of the Subject 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 Perturbation Theory and Molecular Dynamics Definition of the Subject The Framework The Leading Order Born-Oppenheimer Approximation Beyond the Leading Order Future Directions Primary Literature Books and Reviews Quantum Adiabatic Theorem Definition of the Subject and Its Importance 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 Books Quantum Bifurcations Definition of the Subject 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 Convergent Perturbative Expansion in Condensed Matter and Quantum Field Theory 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 Definition of the Subject 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 Perturbation of Equilibria in the Mathematical Theory of Evolution Definition of the Subject 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 Primary Literature Books and Reviews Perturbation Theory for Non-smooth Systems Definition of the Subject 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 Primary Literature Books and Reviews Exact and Perturbation Methods in the Dynamics of Legged Locomotion Definition of the Subject 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) Perturbation Theory for Water Waves Definition of the Subject 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 Periodic Rogue Waves and Perturbation Theory Definition of the Subject 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

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