A Course on Classical Mechanics
Posted by John Baez
Here’s a course on classical mechanics for people who can handle some differential geometry:
 John Baez and Derek Wise, Classical Mechanics, Spring 2005.
This may come in handy if you’re following this Fall’s course on quantization and cohomology  so far we’re mostly covering the same ground at high speed, but soon we’ll shoot ahead.
These are some course notes and homework problems for a mathematics graduate course on classical mechanics. I’ve taught this course twice recently. The first time I focused on the Hamiltonian approach. This time I started with the Lagrangian approach and derived the Hamiltonian approach from that.
My goal was to understand the principle of least action from more viewpoints than I had before, including some historically important ones that you don’t see much these days. I also wanted to show how classical mechanics foreshadows all the ideas of mechanics.
Derek Wise took notes. Here they are:
 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 EulerLagrange 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 specialrelativistic free particle: two Lagrangians, one with reparametrization invariance as a gauge symmetry. 4) A specialrelativistic charged particle in an electromagnetic field.
 A Spring in Imaginary Time. Learn how replacing time by "imaginary time" in Lagrangian mechanics turns dynamics problems involving a point particle into statics problems involving a spring.
 Answers by Garett Leskowitz.
 Answers by Jeffrey Morton.
 Answers by Erin Pearse.
 Answers by Alex Hoffnung.
 Week 4 (Apr. 18, 20, 22)  More example problems: 4) A specialrelativistic charged particle in an electromagnetic field in special relativity, continued. 5) A generalrelativistic 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 positiondependent index of refraction. The ubiquity of geodesic motion. KaluzaKlein 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 HamiltonJacobi equation. Hamilton’s principal function and extended phase space. How the HamiltonJacobi equation foreshadows quantum mechanics.
 Week 8 (May 16, 18, 20)  Towards symplectic geometry. The canonical 1form and the symplectic 2form 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.
You can find errata for these notes here. If you find more errors, please email me!
Here are some homework problems from the last time I taught the course.

The Euclidean Group, the Galilei Group and the Free Particle. Study the symmetries of spacetime in classical physics, and how these act on states of the free particle.
 Answers by Toby Bartels.

Rotations and Angular Momentum.
Study how angular momentum generates rotations in classical
mechanics.
 Answers by Toby Bartels.

The Kepler Problem. Starting from just F = ma, figure out why a body attracted gravitationally by another body must trace out an elliptical, parabolic or hyperbolic orbit.
 Answers by Toby Bartels.

The Kepler Problem Revisited: the RungeLenz vector. Using a special conserved quantity called the RungeLenz vector, solve the Kepler problem much more efficiently.
 Answers by Toby Bartels.

The Pendulum, Elliptic Functions
and Imaginary Time. The motion of a frictionless pendulum is
approximately a sine wave, but the exact solution involves a Jacobi elliptic function. Learn about these, and learn how replacing time by imaginary time explains the double periodicity of this elliptic function!
 Answers by Toby Bartels.
Again, please let me know if you spot errors.
You can also download TeX or LaTeX files of the homework problems and some solutions, if for some bizarre reason you want them. However, the authors keep all rights to this work, except when stated otherwise.
Re: A Course on Classical Mechanics
I’ve looked at some of these notes before and they’re great.
I would like to make some comments about KaluzaKlein theory, which you discussed in the Week 5 lecture.
The first time I learnt of this beautiful fact (that the path an electromagnetic particle follows is the projection of a geodesic on the bundle) was in the excellent book:
I’d like to ask: can one think of string dynamics and gerbes in this way?
Of the many things an (abelian) gerbewithconnection is, I understand that it is at least a linebundlewithconnection on the loop space.
In addition, if your manifold has a metric (as any selfrespecting manifold has), then supposedly this can be lifted to loop space.
So  lets play the KaluzaKlen game… on loop space! We get a metric on the linebundleoverloopspace, and we project the geodesics down to loop space, to obtain dynamics of the loops.
What kind of dynamics do we get out?