Classical Mechanics
John Baez
Here are some course notes and homework problems for a mathematics graduate
course on classical mechanics. There are two versions of the course:
The second course reviews a lot of basic differential geometry.
But, if you'd like to study these courses on your own and
don't feel comfortable with manifolds, vector fields, differential forms
and vector bundles, you might try the following texts, in rough order of
increasing sophistication:

Gregory L. Naber, Topology, Geometry and Gauge Fields:
Foundations, Springer, Berlin, 1997.

Chris Isham, Modern Differential Geometry for Physicists,
World Scientific Press, Singapore, 1999.

John C. Baez and Javier P. Muniain, Gauge Fields, Knots and Gravity,
World Scientific Press, Singapore, 1994. (My favorite, for some
reason. For this class you just need chapters I.2I.4 and II.1II.2.)

Harley Flanders, Differential Forms with Applications to the
Physical Sciences, Dover, New York, 1989. (Everyone has to learn
differential forms eventually, and this is a pretty good place to do it.
Plus, Dover books are cheap!)

Charles Nash and Siddhartha Sen, Topology and Geometry for
Physicists, Academic Press, 1983. (This emphasizes the physics
motivations... it's not quite precise at points.)

Mikio Nakahara, Geometry, Topology, and Physics, A. Hilger, New
York, 1990. (More advanced.)
Everyone should read some books on classical mechanics, too! Here's
the physicist's bible of classical mechanics — a great new version of
the book I used as an undergrad:

Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics,
Addison Wesley, San Francisco, 2002.
And here's a famous book that's closer to the style of this course:

Vladimir I. Arnold, Mathematical Methods of Classical Mechanics,
translated
by K. Vogtmann and A. Weinstein, 2nd edition, Springer, Berlin, 1989.
There are LaTeX and encapsulated Postscript
files of all the material below if
for some bizarre reason you want them. However,
we reserve all rights to these, except for Toby's stuff.
Lagrangian approach
In the Spring of 2005 we started with the Lagrangian approach to classical
mechanics, with a heavy emphasis
on action principles. We then derived the Hamiltonian approach from that.
Derek Wise took
notes, and Blair
Smith converted them into LaTeX, adding extra material and creating
this book:
 John Baez, Blair Smith and Derek Wise, Lectures on Classical
Mechanics.
Available in
PDF and
Postscript.
Here are Derek's original handwritten 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 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.
Homework on 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.
Homework on
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!

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!
In the Winter of 2008 we started with Newton's laws and quickly
headed towards the Hamiltonian approach to classical mechanics,
focusing on Poisson manifolds rather than symplectic manifolds.
Alex Hoffnung created lecture notes in TeX, available below.
These need intensive polishing before they become a book —
or part of a book.
Here are the course notes:

Lecture 1 (Jan. 8) 
A tiny taste of the history of mechanics.
Homework on the falling body and the simple harmonic oscillator.

Answers to homework
by John Huerta.

Answers to homework
by Curtis Pro.

Lecture 2 (Jan. 10) 
Introduction. A classical particle in R^{n}. Momentum and energy.
Conservative forces.

Lecture 3 (Jan. 17) 
A particle in one dimension. A particle in a central force:
angular momentum.
Homework on the
Kepler problem.

Lecture 4 (Jan. 22) 
Many particles in 3 dimensions. Conservation of momentum.
The gravitational nbody problem.
Homework on conservation of energy and
angular momentum for a collection of particles in 3 dimensions
(see lecture notes).

Answers to homework
by Scott Childress.

Answers to homework
by Curtis Pro.

Answers to homework
by Brian Rolle.

Answers to homework
by Michael Maroun.

Lecture 5 (Jan. 25) 
Symmetries and conserved quantities for many particles in
R^{3}. Time translations, spatial translations
and rotations, and Galilei transformations.

Lecture 6 (Jan. 29) 
The center of mass. Symmetries from conserved quantities.
Hamilton's equations.
Homework on the
2body problem and Poisson brackets.

Answers to homework
by Scott Childress.

Answers to homework
by Curtis Pro.

Answers to homework
by Brian Rolle.

Answers to homework
by Michael Maroun.

Lecture 7 (Jan. 31) 
Configuration spaces. Phase spaces. The Poisson algebra
of observables.

Lecture 8 (Feb. 5) 
The phase space for a system whose configuration space
is an arbitrary manifold. Some differential geometry:
tangent and cotangent bundles.

Lecture 9 (Feb. 7) 
Poisson manifolds. The cotangent bundle of R^{n} as a
Poisson manifold. The cotangent bundle of an arbitrary manifold
as a Poisson manifold. Some more differential geometry:
smooth maps, vector fields and 1forms.
Homework on the
Galilei group.

Lecture 10 (Feb. 12) 
How observables generate symmetries. The integral curves of a
vector field. The flow generated by a vector field. The vector
field v_{F} generated by an observable F on a Poisson
manifold.

Lecture 11 (Feb. 14) 
How Hamiltonians generate time evolution: examples. The simple
harmonic oscillator. The particle in a potential in R^{n}.
How observables generate symmetries in the Galilei group: spatial
translations and Galilei boosts.

Lecture 12 (Feb. 19) 
Symmetries and conserved quantities. Definition of conserved
quantity, symmetry. Theorem: an observable G generates symmetries
of an observable F if and only if F generates symmetries of G.
Theorem: any observable F generates symmetries of itself.
So, in particular, energy is automatically conserved in Hamiltonian
mechanics.

Lecture 13 (Feb. 21)  Lie algebras.
The Lie algebra of observables for a Poisson manifold. The Lie algebra
of vector fields for any manifold. How these Lie algebras are related.
Homework on Angular momentum and rotations.

Lecture 14 (Feb. 26) 
The Lie algebra of a Lie group. Actions of Lie groups on manifolds.

Lecture 15 (Feb. 28) 
Group actions preserving structures on manifolds. The structure
on spacetime preserved by the Galilei group (answer to a homework
assigned in lecture 9).

Lecture 16 (Mar. 4) 
Symmetries and observables. How a Lie group acting on a manifold
gives a map from its Lie algebra to the vector fields on this
manifold. The concept of a "Hamiltonian" Lie group action on
a Poisson manifold. Examples, and a famous counterexample.

Lecture 17 (Mar. 6) 
Weakly Hamiltonian group actions. How the concept of "mass"
arises in classical mechanics.
Homework on the LaplaceRungeLenz vector.

Lecture 18 (Mar. 11) 
The category of classical systems. Poisson maps.

Lecture 19 (Mar. 13) 
Cartesian products and noncartesian "tensor products". The
tensor product of Poisson manifolds is noncartesian. So, the
WootersZurek theorem that "you cannot clone a quantum" should
have a classical analogue!
Lagrangian approach: © 2005 John Baez, Derek Wise and Blair Smith
Hamiltonian approach: © 2008 John Baez and Alex Hoffnung
baez@math.removethis.ucr.andthis.edu