Classical Mechanics

John Baez

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.2-I.4 and II.1-II.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.

Lagrangian approach

John Baez, Blair Smith and Derek Wise, Lectures on Classical Mechanics.

Available in PDF and Postscript.

Here are Derek's original hand-written notes:

You can find errata for these notes here. If you find more errors, please email me!

Hamiltonian approach 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.

. 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. Answers to homework by Scott Childress

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Answers to homework by Curtis Pro.

Answers to homework by Brian Rolle.

Answers to an older version of this homework by Toby Bartels.

Lecture 4 (Jan. 22) - Many particles in 3 dimensions. Conservation of momentum. The gravitational n-body 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 R3. Time translations, spatial translations and rotations, and Galilei transformations.

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 Rn as a Poisson manifold. The cotangent bundle of an arbitrary manifold as a Poisson manifold. Some more differential geometry: smooth maps, vector fields and 1-forms. Homework on the Galilei group. Answers to homework by Scott Childress.

Answers to homework by Curtis Pro.

Answers to homework by Michael Maroun.

Answers to homework by Brian Rolle.

Answers to an older version of this homework by Toby Bartels.

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 Rn. 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. Answers to homework by Scott Childress.

Answers to homework by Curtis Pro.

Answers to an older version of this homework by Toby Bartels.

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 Laplace-Runge-Lenz vector. Answers by Scott Childress.

Answers by Curtis Pro.

Answers by Brian Rolle.

Answers to an older version of this homework by Toby Bartels.

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 Wooters-Zurek theorem that "you cannot clone a quantum" should have a classical analogue!