John Baez and Derek Wise
On this page are notes on classical mechanics, written by Derek Wise, based on lectures given by
John Baez during the Spring 2005 quarter at UC Riverside. We are finally working on turning these notes, with additional material, into a proper book on classical mechanics and the associated geometry.
You can find errata for these notes
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
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.
© 2005-2013 John Baez and Derek Wise