Instructor: Ari Stern

Email: stern@wustl.edu

Office: Cupples I, 211B

Office Hours: TuTh 2-3pm

Lectures will be held MWF 12-1pm in Crow 205. The first class will be on Monday, August 27, and the last will be on Friday, December 7. Class will be canceled for Labor Day (Monday, September 3), Fall Break (Monday, October 15), and Thanksgiving Break (Wednesday, November 21, and Friday, November 23).

The primary textbook for this course is *Introduction to
Mechanics and Symmetry*, by Jerrold E. Marsden and Tudor
S. Ratiu (second edition, Springer, 1999). Students enrolled in
this course may use their WUSTL Key login to access
a PDF copy of this textbook.

A secondary text for this course is *Geometric Mechanics and
Symmetry*, by Darryl D. Holm, Tanya Schmah, and Cristina
Stoica (Oxford,
2009). A PDF
copy of this textbook is available from Prof. Holm's
website.

Three additional supplemental texts on geometric mechanics
are: *Foundations of Mechanics*, by Ralph Abraham and
Jerrold E. Marsden; *Mathematical Methods of Classical
Mechanics*, by V. I. Arnold; and *Global Formulations of
Lagrangian and Hamiltonian Dynamics on Manifolds*, by Taeyoung
Lee, Melvin Leok, and N. Harris McClamroch.

All five of these texts have been placed on reserve at Olin Library.

On each of the following due dates, hand in solutions to five problems of your choosing, taken from the Marsden-Ratiu and/or Holm-Schmah-Stoica textbooks:

- HW1: Due Friday, September 7.
- HW2: Due Friday, September 21.
- HW3: Due Friday, October 5. (Final project topic description also due.)
- HW4: Due Friday, October 19.
- HW5: Due Friday, November 2.
- HW6: Due Friday, November 16. (Final project summary/outline also due.)

In lieu of exams, each student will give a 20-minute final
presentation on a topic in geometric mechanics *not*
covered in the Marsden-Ratiu textbook. The possibilities are
almost endless, but here are a few ideas to get you started:
systems with nonholonomic constraints, presymplectic Lagrangian
systems and the Gotay-Nester algorithm, Dirac structures and the
Hamilton-Pontryagin variational principle, the EPDiff equation,
multisymplectic geometry and Lagrangian/Hamiltonian field
theory, discrete-time geometric mechanics and symplectic/Poisson
numerical integrators, mechanics on Lie algebroids,
“port-Hamiltonian” systems, geometric quantization,
etc. I encourage you to discuss potential choices of topics with
me, and I will be happy to point you towards some of the
relevant literature to get you started.

The deliverables and due dates for the project are as follows:

- Friday, October 5: Brief description of the topic you have chosen, how you plan to approach it, and the references you plan to use. (1 page, typed in LaTeX)
- Friday, November 16: Summary of your project, with a detailed outline of your upcoming presentation. (2-3 pages, typed in LaTeX)
- Last week of classes (December 3-7): Final presentations. Hand in a copy of your lecture notes (if a board talk) or slides (if a slide talk).

Grades will be based on a weighted average of homework (20%), final project description (10%), final project summary (20%), and final presentation (50%).

A graduate-level introduction to classical mechanics from the
modern, differential geometric point of view. Topics include: the
Lagrangian and Hamiltonian formalisms, symplectic and Poisson
geometry, Lie groups and Lie algebras, symmetries, conservation
laws, reduction. *Prerequisites*: prior exposure to manifolds and to
groups or permission of the instructor.