Nonlinear Mechanics - Physics at Oregon State University

Contents
1 Lagrangian Dynamics 5
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Generalized Coordinates and the Lagrangian . . . . . . . . . 6
1.3 Virtual Work and Generalized Force . . . . . . . . . . . . . . 8
1.4 Conservative Forces and the Lagrangian . . . . . . . . . . . . 10
1.4.1 The Central Force Problem in a Plane . . . . . . . . . 11
1.5 The Hamiltonian Formulation . . . . . . . . . . . . . . . . . . 13
1.5.1 The Spherical Pendulum . . . . . . . . . . . . . . . . . 15
2 Canonical Transformations 17
2.1 Contact Transformations . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 The Harmonic Oscillator: Cracking a Peanut with a
Sledgehammer . . . . . . . . . . . . . . . . . . . . . . 20
2.2 The Second Generating Function . . . . . . . . . . . . . . . . 21
2.3 Hamilton’s Principle Function . . . . . . . . . . . . . . . . . . 22
2.3.1 The Harmonic Oscillator: Again . . . . . . . . . . . . 24
2.4 Hamilton’s Characteristic Function . . . . . . . . . . . . . . . 25
2.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Action-Angle Variables . . . . . . . . . . . . . . . . . . . . . . 27
2.5.1 The harmonic oscillator (for the last time) . . . . . . . 29
3 Abstract Transformation Theory 33
3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . 35
3.2 Geometry in n Dimensions: The Hairy Ball . . . . . . . . . . 38
3.2.1 Example: Uncoupled Oscillators . . . . . . . . . . . . 41
3.2.2 Example: A Particle in a Box . . . . . . . . . . . . . . 43
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