Hamiltonian Structures in Perturbation Theory

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