We will study the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. We will use computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration.

We will consider the following topics: The Lagrangian formulation. Action, variational principles, and equations of motion. Hamilton's principle. Conserved quantities. Rigid bodies and tops. Hamiltonian formulation and canonical equations. Surfaces of section. Chaos. Canonical transformations and generating functions. Liouville's theorem and Poincaré integral invariants. Poincaré-Birkhoff and KAM theorems. Invariant curves and cantori. Nonlinear resonances. Resonance overlap and transition to chaos. Properties of chaotic motion.

Ideas will be illustrated and supported with physical examples. We will make extensive use of computing to capture methods, for simulation, and for symbolic analysis.

This subject awards H-LEVEL Graduate Credit, however the subject is appropriate for undergraduates who have taken the prerequisites. Undergraduates are welcome.

Permission of instructors required.

Calendar

Introduction

Lagrangian Mechanics

Rigid Bodies

Hamiltonian Mechanics

Phase Space Structure

Canonical Transformations