Joint Subject Offering: 6.946J, 8.351J, 12.620J

**Classical Mechanics:
A Computational Approach**

Jack Wisdom, 54-414, x3-7730

Gerald Jay Sussman, 32G-514, x3-5874

Kenny Chen will assist us this term.

**
Content:
**

We study the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. We use computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration.

Our presentation is unusual in that computation underlies our understanding of the mathematical ideas underlying mechanics. We make our mathematical notations explicit and precise enough that they can be interpreted automatically by a computer. As a consequence of this requirement the formulas and equations stand on their own. They have clear meaning, independent of any informal context.

We plan to consider the following topics: The Lagrangian formulation. Action, variational principles, and equations of motion. Hamilton's principle. Conserved quantities. Hamiltonian formulation and canonical equations. Surfaces of section. Chaos. Canonical transformations and generating functions. Hamilton-Jacobi theory, Liouville's theorem and Poincaré integral invariants. Canonical perturbation theory. Poincaré-Birkhoff and KAM theorems. Invariant curves. Nonlinear resonances. Resonance overlap and transition to chaos. Properties of chaotic motion.

Ideas will be illustrated and supported with physical examples. We will make extensive use of computing to capture methods, for simulation, and for symbolic analysis.

This subject awards G-LEVEL Graduate Credit, however the subject is appropriate for undergraduates who have the prerequisite experience. Undergraduates are welcome.

Prerequisites: 8.01, 18.03, programming experience

Lectures: MWF at 1400EDT (2:00PM) Online.

Computer Lab: Wednesday evenings, 1900-2200 EDT (7--10 PM).

Units: 3-3-6

**
Teaching plan:
**

Professors Jack Wisdom (EAPS) and Gerald Jay Sussman will teach this class entirely online, with synchronous lectures on Monday, Wednesday, and Friday at 1400--1500 EDT (2:00PM--3:00PM). We will run our evening laboratory, each Wednesday from 1900--2200 EDT (7:00PM--10:00PM). We expect students to actively participate in all of the meetings, so it may be difficult for students who are in some time zones. If a significant number of students want the hours to be modified, we will be happy to adjust, as appropriate, (one advantage of online teaching is that there is not a problem with allocating a classroom!) The times can be renegotiated by the class in the first few meetings of the term.

We also invite students individually and in groups to drop in (online) to discuss problems with us at other times. We will try to make ourselves very accessible in these strange circumstances.

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Grades:
**

This is an advanced class. There are no quizzes. Grades are entirely determined by the work that students do in weekly problem sets, participation in the laboratory, and some more extensive projects. In the evening laboratory we hang out with the students, engage in discussions, and we try to help with the weekly work. We encourage students to work together and help each other learn the material, but each student must prepare an individual report on the work done on the problem sets and projects. Each student must report on the help that they gave to and received from other students.

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Textbook:
**

Structure and Interpretation of Classical Mechanics, second edition, MIT Press, 2014, ISBN: 978-0-262-02896-7

Mechanics Book (HTML), second edition

Errata for the Mechanics Book, second edition .

Structure and Interpretation of Computer Programs, second edition (HTML)

If you want to install the software on your personal computers see here.

We will also make our software available on Athena.

Because our laboratory will be online this year (UGH!) we are also developing a special collaborative software system that we will use in the lab!

Our MIT Press book on differential geometry!

**
Software for Differential Geometry
**
This software is now automatically available as part of our mechanics system,
if you get the latest version here.

**
Experimental Video Presentations
**
here.