Classical Mechanics

John Baez and Derek Wise

Spring 2005

These notes on classical mechanics were taken by Derek Wise, based on lectures given by John Baez during the Spring 2005 quarter at UCR. We have dreamed of turning these notes, along with notes from the previous time John taught this course, into a book eventually.... but it will be a while before that happens!

• Week 1  (Mar. 28, 30, Apr. 1) - The Lagrangian approach to classical mechanics: deriving F = ma from the requirement that the particle's path be a critical point of the action. The prehistory of the Lagrangian approach: D'Alembert's "principle of least energy" in statics, Fermat's "principle of least time" in optics, and how D'Alembert generalized his principle from statics to dynamics using the concept of "inertia force".
• Week 2  (Apr. 4, 6, 8) - Deriving the Euler-Lagrange equations for a particle on an arbitrary manifold. Generalized momentum and force. Noether's theorem on conserved quantities coming from symmetries. Examples of conserved quantities: energy, momentum and angular momentum.
• Week 3  (Apr. 11, 13, 15) - Example problems: 1) The Atwood machine. 2) A frictionless mass on a table attached to a string threaded through a hole in the table, with a mass hanging on the string. 3) A special-relativistic free particle: two Lagrangians, one with reparametrization invariance as a gauge symmetry. 4) A special-relativistic charged particle in an electromagnetic field.
• Week 4  (Apr. 18, 20, 22) - More example problems: 4) A special-relativistic charged particle in an electromagnetic field in special relativity, continued. 5) A general-relativistic free particle.
• Week 5  (Apr. 25, 27, 29) - How Jacobi unified Fermat's principle of least time and Lagrange's principle of least action by seeing the classical mechanics of a particle in a potential as a special case of optics with a position-dependent index of refraction. The ubiquity of geodesic motion. Kaluza-Klein theory. From Lagrangians to Hamiltonians.
• Week 6  (May 2, 4, 6) - From Lagrangians to Hamiltonians, continued. Regular and strongly regular Lagrangians. The cotangent bundle as phase space. Hamilton's equations. Getting Hamilton's equations directly from a least action principle.
• Week 7  (May 9, 11, 13) - Waves versus particles: the Hamilton-Jacobi equation. Hamilton's principal function and extended phase space. How the Hamilton-Jacobi equation foreshadows quantum mechanics.
• Week 8  (May 16, 18, 20) - Towards symplectic geometry. The canonical 1-form and the symplectic 2-form on the cotangent bundle. Hamilton's equations on a symplectic manifold. Darboux's theorem.
• Week 9  (May 23, 25, 27) - Poisson brackets. The Schrödinger picture versus the Heisenberg picture in classical mechanics. The Hamiltonian version of Noether's theorem. Poisson algebras and Poisson manifolds. A Poisson manifold that is not symplectic. Liouville's theorem. Weil's formula.
• Week 10  (June 1, 3, 5) - A taste of geometric quantization. Kähler manifolds.

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