Geometric Mechanics
Classical mechanics studies the motion of mechanical systems. It originates in celestial mechanics (which remains an active area).
The geometric aspect of mechanics arises in several ways. First, mechanical systems are generally constrained, meaning the set of all possible configurations is usually a smooth manifold. Kinetic energy in this case is generally defined by some natural Riemannian metric. Second, these mechanical systems are frequently conservative, meaning that energy is conserved, so the systems are tangent to the iso-energy manifolds. Third, there are usually variational structures underlying these systems. Finally, the ``dual'' to the variational structure is a natural geometric structure called a symplectic structure.
Objectives
We will establish the foundations of modern mechanics from the Lagrangian and Hamiltonian points-of-view. We will then learn about symmetry groups and complete integrability, and then Mather theory. If we have time, we will briefly touch on the fascinating topic of nonholonomic mechanics.
Outline
We will examine the following topics in this course:
| Topic | Subject | Weeks |
|---|---|---|
| Introduction | Classical mechanics & differential geometry | 1-2 |
| Chapter 1 | The Lagrangian Formalism | 3-4 |
| Chapter 2 | The Hamiltonian Formalism | 4-5 |
| Chapter 3 | Symmetry | 5-6 |
| Chapter 3 | Complete integrability | 7-10 |
| Chapter 4 | Mather Theory | 11-14 |
| Chapter 5 | Nonholonomic mechanics | 15 |
| - | Presentations | 16. |
Evaluation
Your final grade will be determined by the following:
| Description | Contribution |
|---|---|
| Weekly assignments | 40% |
| Individual Presentation | 20% |
| Final Exam | 40%. |
- The weekly assignments will be based on the lectures and associated readings.
- Presentation topics will be available early in the semester.
- The final exam will take place in Moore 107 at 1400—1550 (2:00—3:50pm) on Monday, April 30, 2012. It will last 1 hour 50 minutes. Exam schedules.
Assignments
The weekly assignments. To view the html pages, you need to enable javascript.
| Week | Assignment | Solution | Week | Assignment | Solution |
|---|---|---|---|---|---|
| 1 | pdf/html | pdf/html | 8 | pdf/html | pdf/html |
| 2 | pdf/html | pdf/html | 9 | pdf/html | pdf/html |
| 3 | pdf/html | pdf/html | 10 | pdf/html | pdf/html |
| 4 | pdf/html | pdf/html | 11 | pdf/html | pdf/html |
| 5 | pdf/html | pdf/html | 12 | pdf/html | pdf/html |
| 6 | pdf/html | pdf/html | 13 | pdf/html | pdf/html |
| 7 | pdf/html | pdf/html | 14 | pdf/html | pdf/html |
| 15 | pdf/html | pdf/html |
Tutorial Questions
| Week | Tutorial | Solution |
|---|---|---|
| 9 | pdf/html | pdf/html |
| 10 | pdf/html | pdf/html |
| 11 | pdf/html | pdf/html |
| 12 | pdf/html | pdf/html |
| 13 | pdf/html | pdf/html |
| 14 | pdf/html | pdf/html |
Time & Location
| Week | Date | Day | Time | Location | |
|---|---|---|---|---|---|
| 1 | 13 Jan | Friday | 1600-1650 | PE 225 | |
| 2 | 17 Jan | Tuesday | 1600-1750 | PE 107 | continues week 1 lecture |
| 2 | 20 Jan | Friday | 1400-1650 | PE 225 | |
| 3 | TBA | TBA | TBA | TBA | |
| Otherwise | * | Friday | 1400-1650 | PE 225 |
Lecture Notes
| Week | Notes | Week | Notes |
|---|---|---|---|
| 1 | pdf/html | 8 | pdf/html |
| 2 | pdf/html | 9 | pdf/html |
| 3 | pdf/html | 10 | pdf/html |
| 4 | pdf/html | 11 | pdf/html |
| 5 | pdf/html | 12 | pdf/html |
| 6 | pdf/html | 13 | pdf/html |
| 7 | pdf/html | 14 | pdf/html |
| 15 | presentations |
Presentation Topics
| Topic | Reference(s) | Presenter |
|---|---|---|
| Motion in a Planar Central Force | Arnold, chapter 2, section 8 | AA |
| Rigid Bodies | Arnold, chapter 6, sections 26-28 | SW |
| Normal Forms of Quadratic Hamiltonians | Arnold, Appendices 6-7 | CL |
| Geodesics on the 3-sphere | Bates and Cushman, chapter 2, sections 1-2 | TP |
| Regularization of the Kepler Problem | Bates and Cushman, chapter 2, section 3.4 |
The presentations will be done in the final week of classes, Wednesday, 25 April from 5-7 pm in Pearce 203. Here is a sample gradesheet.
Sources
We will draw from a variety of sources, including:
@preamble{ "\def\cprime{$'$} " } @book {MR1345386, AUTHOR = {Arnol{\cprime}d, V. I.}, TITLE = {Mathematical methods of classical mechanics}, SERIES = {Graduate Texts in Mathematics}, VOLUME = {60}, NOTE = {Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {199?}, PAGES = {xvi+516}, ISBN = {0-387-96890-3}, MRCLASS = {70-02 (58F05 58Fxx 70Hxx)}, MRNUMBER = {1345386 (96c:70001)}, } @book {MR1438060, AUTHOR = {Cushman, Richard H. and Bates, Larry M.}, TITLE = {Global aspects of classical integrable systems}, PUBLISHER = {Birkh\"auser Verlag}, ADDRESS = {Basel}, YEAR = {1997}, PAGES = {xvi+435}, ISBN = {3-7643-5485-2}, MRCLASS = {58F07 (58-01 70E15 70F05 70H05)}, MRNUMBER = {1438060 (98a:58083)}, MRREVIEWER = {Lubomir Gavrilov}, } @book {MR1066693, AUTHOR = {Guillemin, Victor and Sternberg, Shlomo}, TITLE = {Symplectic techniques in physics}, EDITION = {Second}, PUBLISHER = {Cambridge University Press}, ADDRESS = {Cambridge}, YEAR = {1990}, PAGES = {xii+468}, ISBN = {0-521-38990-9}, MRCLASS = {58F05 (58-02 58F06 70Hxx)}, MRNUMBER = {1066693 (91d:58073)}, } # end of index.bib
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| Spherical Pendulum. |
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