Math 547

Geometric Mechanics, Fall 2018

Basic Information

Instructor: Ari Stern
Email: stern@wustl.edu
Office: Cupples I, 211B
Office Hours: TuTh 2-3pm

Lectures

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).

Textbook and Supplemental Texts

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.

Homework Assignments

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.)

Final Project

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).

Grading

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

Catalog Description

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.