Lectures on classical mechanics

Baez J., Smith B., Wise D.

These are notes for a mathematics graduate course on classical mechanics. I’ve taught this course twice recently. The
rst time I focused on the Hamiltonian approach. This time I started with the Lagrangian approach, with a heavy emphasis on action principles, and derived the Hamiltonian approach from that. Derek Wise took notes.The chapters in this LATEX version are in the same order as the weekly lectures, but I’ve merged weeks together, and sometimes split them over chapter, to obtain a more textbook feel to these notes. – John C. Baez

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Table of contents :
Lagrangian and Newtonian Approaches……Page 8
Prehistory of the Lagrangian Approach……Page 12
The Principle of Minimum Energy……Page 13
D’Alembert’s Principle and Lagrange’s Equations……Page 14
The Principle of Least Time……Page 16
How D’Alembert and Others Got to the Truth……Page 18
The Euler-Lagrange Equations……Page 20
Lagrangian Dynamics……Page 22
Interpretation of Terms……Page 24
Time Translation……Page 26
Canonical and Generalized Coordinates……Page 27
Symmetry and Noether’s Theorem……Page 28
Noether’s Theorem……Page 29
Conserved Quantities from Symmetries……Page 30
Space Translation Symmetry……Page 31
Rotational Symmetry……Page 32
The Atwood Machine……Page 34
Disk Pulled by Falling Mass……Page 35
Free Particle in Special Relativity……Page 37
Gauge Symmetry and Relativistic Hamiltonian……Page 41
Relativistic Hamiltonian……Page 42
Relativistic Particle in an Electromagnetic Field……Page 43
Lagrangian for a String……Page 45
Alternate Lagrangian for Relativistic Electrodynamics……Page 47
The General Relativistic Particle……Page 49
Free Particle Lagrangian in GR……Page 50
Charged particle in EM Field in GR……Page 51
Jacobi and Least Time vs Least Action……Page 52
The Ubiquity of Geodesic Motion……Page 54
The Hamiltonian Approach……Page 57
Example: A Particle in a Riemannian Manifold with Potential V(q)……Page 60
Example: A Regular but not Strongly Regular Lagrangian……Page 61
Hamilton’s Equations……Page 62
Hamilton and Euler-Lagrange……Page 63
Hamilton’s Equations from the Principle of Least Action……Page 65
Wave Equations……Page 67
The Hamilton-Jacobi Equations……Page 70