Survey of Geometric Perturbation Theory
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
