Geometric Perturbation Theory in Physics

The following sections are included:

• Preface

• Acknowledgements

• Table of Contents

The following sections are included:

• Summary and Motivation

• Philosophical Approach

• Guide for the Reader

• Intuitive Discussion of the Conceptual Framework
  • The Analogy Between Entropy and Action
  • Adiabatic Invariants and Pseudo-forces
  • Symmetries and Exact Invariants
  • Thermodynamic Forces
  • Wave Action
  • Action, Entropy and Asymptotics
  • Steepest Descents, Stationary Phase, and Averaging
  • The Key Examples in Mechanics, Waves, and Statistics
  • Mechanical Systems
  • Mechanical Systems: Separation of Time Scales
  • Averaging in Statistical Systems
  • Averaging and Local Fourier Transforms
  • Symplectic Asymptotics in Thermodynamics and Mechanics

• A Hundred Further Questions, Conjectures, and Suggestions

The following sections are included:

• Historical Background

• Geometric Perturbation Theory
  • Manifolds
  • Dynamical Systems
  • Perturbation Theory
  • First Order Perturbation Equations
  • Functions, Covectors, and Cotangent Bundles
  • Vectors and Tangent Bundles
  • The State Space for First Order Perturbation Theory
  • Flows and Derivatives
  • Dynamics for First Order Perturbation Theory

• The Geometry of Jth Order Perturbation Theory
  • The Path Space
  • The Space of Germs of Paths
  • The Space of Jets of Paths
  • Tangent Vectors to Path Space
  • Tangent Vectors to the Quotient Spaces
  • Dynamics on Path Space
  • Dynamics on Jet Space

• Geometric Hamiltonian Mechanics
  • Poisson Manifolds
  • Hamiltonians and Hamiltonian Vector Fields
  • Symplectic Manifolds
  • Symplectic Leaves and Bones and Casimir Functions
  • The Natural Symplectic Structure on Cotangent Spaces

• Hamiltonian Systems with Symmetry
  • Generalized Noether’s Theorem
  • Circle Actions
  • Reduction by a Circle Action

• Example: Centrifugal Force
  • Angular Momentum Generates Rotations
  • The Reduced Space and Bracket
  • The Reduced Hamiltonian Gives Centrifugal Force

• Higher Dimensional Symmetries
  • Hamiltonian Symmetry
  • The Momentum Map
  • Non-commutativity as the Obstruction to Reduction
  • The Adjoint and Coadjoint Actions
  • Multidimensional Reduction using a Coadjoint Isotropy Subgroup
  • Multidimensional Reduction using Coadjoint Orbits
  • The Lie-Poisson Bracket and Group Configuration Spaces
  • Euler’s Equations for the Free Rigid Body
  • Euler’s Equations for a Perfect Fluid
  • Gases and Plasmas

• Geometric Hamiltonian Perturbation Theory
  • Linearized Dynamics at a Fixed Point from Jet Bracket
  • Symmetry and Perturbation Theory

• The Method of Averaging for Hamiltonian Systems
  • Approximate Noether’s Theorem
  • Hamiltonian Averaging as Reduction by a Circle Action
  • Pseudo-Potentials and Adiabatic Invariants

• Example: E × B Drift

The following sections are included:

• Pseudo-forces and Reduction
  • Fictitious Forces
  • Rotating Coordinates
  • Reduction of Simple Mechanical Systems
  • Circle Actions on Simple Mechanical Systems
  • Fictitious Forces as Reduction
  • Fictitious Forces with Momentum Shift as Reduction

• Geometry of the Method of Variation of Parameters
  • Hamiltonian Variation of Parameters

The following sections are included:

• Introduction

• First Order Hamiltonian Perturbation Theory

• Path and Jet Spaces
  • Path Space Symplectic Structure and Hamiltonian
  • The Path Space Dynamics is Hamiltonian

• Coordinate Description of the J-jet Structure
  • The Jet Hamiltonian
  • The Jet Poisson Bracket

• Relation to the Iterated Tangent Bundle
  • Injecting Jets into the Iterated Tangent Bundle
  • Symplectic Structure on the Iterated Tangent Bundle
  • Pulled Back Symplectic Structure on the Jet Space

• Relation to the Path Space Bracket
  • Weighted Path Bracket and Hamiltonian
  • Jet Bracket Arises from Derivative of Delta Function Weighting
  • Jet Hamiltonian from Derivative of Delta Function Weighting

• Jet Space as Derivative
  • The Sheet Quotient Spaces
  • Sheet Symplectic Structure and Hamiltonian
  • Map Between Sheet Space and Jet Space
  • The Pulled Back Sheet Symplectic Structure and Hamiltonian
  • Sheet Structures Asymptote to Jet Structures for Small Spacing

• Jets and Symmetry
  • ε-dependent Group Actions on M
  • The Path Group: PG
  • The Jet Group: JG
  • When M is a Coadjoint Orbit with the KKS Symplectic Structure
  • Natural Projections and Injections of G, PG, and JG
  • The Lie Frisson Bracket on g*
  • JG as a Semi-Direct Product
  • Jet and Path Reduced Spaces are Reduced Jet and Path Spaces

The following sections are included:

• Introduction: Kruskal’s Approach

• The New Approach Expressed in Coordinates

• The Geometric Version of the New Approach

• Extensions and Limitations of Kruskal’s Theory
  • Secular Perturbation Theory
  • 1/ε Time of Validity for Kruskal’s Technique
  • Averaging with Multiple Frequencies
  • Averaging Over Ergodic Orbits
  • Non-uniqueness of Symmetry for Finite Perturbation

The following sections are included:

• Ponderomotive-like Forces
  • Intuitive Treatment
  • Introducing the Asymptotic Scaling
  • Using the Kruskal-like Perturbation Method
  • Obtaining the Action to First Order

• Some Comments on Perturbation Calculations

• 2D Gyromotion via the New Kruskal-like Method
  • The Four Components of R₁
  • The Four Components of R₂
  • Summary of the Calculation

• The Hamiltonian Structure of Gyromotion
  • The Poisson Bracket
  • The Symplectic Structure
  • Uniqueness of the Generator for a Vector Field
  • Comparison With Robert Littlejohn’s Results
  • Prospects for a Fully Hamiltonian Theory

The following sections are included:

• Wave Asymptotics and Approximate Symmetry
  • Eikonal Waves
  • The Local Fourier Transform
  • Stationary Phase, Laplace’s Method, and Steepest Descents
  • Eikonal Waves and Lagrangian Submanifolds

• WKB Theory and Asymptotic Equations
  • The Structure of First Order P.D.E.’s
  • Hamilton-Jacobi Theory and Symplectic Manifolds
  • Cotangent Bundles, Contact Spaces, and Jet Spaces
  • The Contact Bundle and the Conormal Bundle

• Limitations of WKB Theory

The following sections are included:

• Introduction

• Periodic Solutions
  • The Hamiltonian Restricted to Periodic Solutions
  • The Symplectic Structure Restricted to Periodic Solutions
  • The Action of Periodic Orbits

• Modulations
  • Stationary Phase Integrals
  • The Modulational Poisson Brackets
  • The Modulational Hamiltonian

• Global Symmetry Implies Local Conservation Law

• The Nonlinear Klein-Gordon Equation

The following sections are included:

• Explicit Calculation of the Lie Poisson Bracket

• The Geometrical Picture

The following sections are included:

• Imbedding in a Hamiltonian System

• Projection from a Hamiltonian System

• Dangerous Operations with Unbounded Variables
  • Eg.: Surreptitiously Changing Damping to Driving
  • Eg.: Pitfalls in the Use of Lie Transforms

• Imbedding in Poisson Systems

The following sections are included:

• Application to Particle Accelerators
  • Courant’s Theorem

• Relation to the Uncertainty Principle

• Weinstein’s Approach

• Theorem for Linear Canonical Transformations
  • The Geometry of Projected Parallelepipeds
  • The Case of Linear Canonical Transformations

The seeming paradox of reversible underlying dynamics leading to irreversible macroscopic behavior has been wrestled with since the time of Boltzmann. For the case of gas dynamics, Boltzmann derived his famous equation on the basis of the questionable statistical assumption of “Stosszahlansatz”. From the Boltzmann equation one may derive the “H theorem” which says that the time derivative of the entropy is greater than or equal to zero. The “Stosszahlansatz” or “molecular chaos assumption” says that the probability distributions for colliding molecules should be uncorrelated. As has been pointed out many times (see for example: pp. 46-88 of [Chapman and Cowling, 1958] and pp. 28-32 of the Statistical Physics volume in [Landau and Lifshitz, 1960-1981]), because the underlying dynamics is reversible, for every state with its entropy increasing, there is a corresponding state with its entropy decreasing. In fact, one may show from this argument that the stosszahlansatz can hold only when the time derivative of entropy is zero. There is no intrinsically special direction in time: If one starts with a random state with low entropy, the entropy increases if one follows the evolution of the state either backwards or forwards in time. There are many more states with high entropy than low, and so a system is likely to be in a high entropy state regardless of the time…

The following sections are included:

• Poincaré Recurrence and Attractors
  • The Lack of Recurrence in Infinite Dimensions

• Asymptotic and Liapunov Stability
  • Almost Attractors

• Reversible Attractors and Infinite Dimensional Hamiltonians
  • Reversible Almost Attractor in the Wave Equation
  • The Liouville Equation and Koopmanism
  • Landau Damping
  • The Boltzmann Equation and the BBGKY Hierarchy
  • Dissipation from Resonance
  • Resonant Coupling of Eikonal Waves

• The String with a Spring

The following sections are included:

• The 2x mod 1 Map

• The Baker’s Transformation

• The Horseshoe

• Example of Horseshoes in Gyromotion

• Area Preserving Horseshoes and Reinsertion

The following sections are included:

• Scaling and Universality

• Magnetic Spin Lattices

• The Central Limit Theorem

• A Poor Man’s Feigenbaum Number

The following sections are included:

• Previous Approaches to Geometric Thermodynamics

• Seven Approaches to the Maximum Entropy Formalism
  • Axiomatic Subjective Approach
  • Counting Sequences of Trials
  • Via Steepest Descents in Two Ways
  • Via Probability in Three Ways

• The Thermodynamic Limit
  • The Density of States
  • The Partition Function

• Maximum Entropy Applied to Statistical Mechanics

• Some Symplectic and Contact Geometry
  • Hypersurfaces Determined by a Function
  • The Conormal Bundle
  • The Wavefront Set
  • The Space of Tangent Contact Elements
  • Legendre Transforms and Linear State Spaces

• The Origin of the Lagrangian Submanifolds in Physics
  • Constrained Integration and Extremization
  • Paths Constrained on Surfaces
  • The Wavevector as a Kind of Force
  • Distributions Constrained on Subsystems
  • Thermodynamic Forces
  • Lagrange Multipliers and Legendre Maps
  • Lagrangian Submanifolds and Constrained Extremization

• Theorem on the Pushforward of Legendre Submanifolds
  • The Contact Structure for Thermodynamics
  • Legendre Transforms and Thermodynamic Potentials

• Phase Transitions and the Geometry of the Equation of State
  • Caustics and Phase Transitions
  • Convexity and First Order Phase Transitions
  • A Generalization of Maxwell’s Equal Area Rule

• Relations Between Symplectic Thermodynamics and Mechanics
  • A) Eikonal Waves and Stationary Phase
  • B) Thermodynamic Limit and Steepest Descents
  • A) Waves and the Feynman Path Integral
  • B) Probability and the Maximum Entropy Formalism
  • A) Wave Path Integrals over a Subspace
  • B) Probability Distribution Averages over a Subspace
  • A) Lagrange Multipliers and Canonical Conjugacy
  • B) Lagrange Multipliers and Thermodynamic Conjugacy
  • A) Fourier Transforms and Legendre Transforms
  • B) Laplace Transforms and Legendre Transforms

The following sections are included:

• Glossary of Terms

• Bibliography