edited by S M Omohundro (UC, Berkeley)

This book which focusses on mechanics, waves and statistics, describes recent developments in the application of differential geometry, particularly symplectic geometry, to the foundations of broad areas of physics. Throughout the book, intuitive descriptions and diagrams are used to elucidate the mathematical theory. It develops a coordinate-free framework for perturbation theory and uses this to show how underlying symplectic structures arise from physical asymptotes. It describes a remarkable parity between classical mechanics which arises asymptotically from quantum mechanics and classical thermodynamics which arises asymptotically from statistical mechanics. Included here is a section with one hundred unanswered questions for further research.

• Introduction
• Survey of Geometric Perturbation Theory. Pseudo-Forces and Reduction
• Hamiltonian Structures in Perturbation Theory
• Kruskal's Theory of Nearly Periodic Systems
• Ponderomotive Force and Gyromotion
• Asymptotic Wave Theory
• A Hamiltonian Approach to Wave Modulation
• A Lie-Poisson Bracket for Wave Action Density
• Imbedding and Projection Theorems
• Projected Area and Canonical Transformations
• Reversibility vs. Irreversibility
• Hamiltonian Dissipation in Infinite Dimensions
• Reinsertion in Area-Preserving Horseshoes
• Renormalization Group
• Symplectic Thermodynamics from Maximum Entropy