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MATH 277A Geometric Mechanics

MWF 11:00-11:50am, APM 2402 (Moved from APM 7421).

Instructor

Prof. Melvin Leok
Office: APM 5763
Email: mleok@math.ucsd.edu
Office Hours: MW 1:00pm-1:50pm, or by appointment.

Announcements

Course Handout
The resources below are password protected with the user name ma277a, and the password is the first 4 digits of:
$\sqrt{6\sum_{n=1}^\infty}{1}{n^2}}$ .

Homework

As there is no TA for this class, there will be no assessed homework.
However, Prof. Darryl Holm, the author of the textbook, has a good collection of homework problems, and solutions: Homework Problems and Solutions
You are encouraged to look over these problems, and try to work out some of these for yourself. If you would like suggestions for which ones to concentrate on, please contact Prof. Leok.

Final Presentation Schedule

Monday, March 19, 2012

11:30am - 11:50am	Maximilian Metti	Finite-Element Exterior Calculus
11:50am - 12:10pm	John Moody	Symplectic and Multi-symplectic Wave Equation
12:10pm - 12:30pm	Helen Parks	Euler-Poincare reduction in the case of continuum fluid motion
12:30pm - 12:50pm	James Tian	Hamiltonian Geometrodynamics
12:50pm - 1:10pm	Chris Tiee	On the Nature of Canonical One-forms

Friday, March 23, 2012

8:00am - 8:20am	James Aisenberg	Variational Principle for the Schrodinger Equation
8:40am - 9:00am	Yawo Ezunkpe	Point Vortices
9:00am - 9:20am	Angelo Scandaliato	Uncertainty propagation on SO(3)
9:20am - 9:40am	Andrew McLeod	Connections, Curvature, and General Relativity
9:40am - 10:00am	Yury Kiselev	Gauge Field Theories
10:00am - 10:20am	Joe Salamon	An Introduction to the History and Applications of Noncommutative Geometry

Course Description

This course will introduce geometric mechanics at the graduate level, which involves the use of geometric and symmetry techniques in the analysis of mechanical systems. In particular, we will discuss the variational principles and geometric structures that underly the Lagrangian and Hamiltonian formulation of mechanics, as well as introducing the relevant tools of differential geometry, such as manifolds, exterior calculus, and Lie groups. The course will culminate in a discussion of how symmetry and reduction theory serve as a unified basis for understanding the Eulerian description of rigid body dynamics and fluid mechanics.

Background

Some exposure to analytical mechanics is helpful, but not essential. The course will introduce the relevant differential geometric tools as well as the relevant aspects of analytical mechanics.

Textbook

Darryl Holm, Geometric Mechanics - Part I: Dynamics and Symmetry, Second Edition, Imperial College Press, 2011. ISBN: 184816775X. [ Electronic Version ]

Additional Reading

You should consider these additional references as a potential source of advanced topics in geometric mechanics, which would be suitable for further study, as part of your course project.

Ralph Abraham, Jerrold Marsden, Foundations of Mechanics, Second Edition, American Mathematical Society 2008. ISBN: 0821844385. [ Electronic Version ]
Ralph Abraham, Jerrold Marsden, Tudor Ratiu, Manifolds, Tensor Analysis, and Applications, Second Edition, Springer-Verlag 1988. ISBN: 0387967907. [ Draft of Third Edition ]
Vladimir Arnold, Mathematical Methods of Classical Mechanics, Second Edition, Springer-Verlag 1989. ISBN: 0387968903. [ Electronic Version ]
Anthony Bloch, Nonholonomic Mechanics and Control, Springer-Verlag, 2010. ISBN: 1441930434. [ Electronic Version ]
Theodore Frankel, The Geometry of Physics, Second Edition, Cambridge University Press, 2003. ISBN: 0521539277. [ Electronic Version ]
Darryl Holm, Geometric Mechanics - Part II: Rotating, Translating and Rolling, Second Edition, Imperial College Press, 2011. ISBN: 1848167784. [ First Edition ]
Darryl Holm, Tanya Schmah, Cristina Stoica, Geometric Mechanics and Symmetry, Oxford University Press, 2009. ISBN: 0199212910. [ Electronic Version ]
Chris Isham, Modern Differential Geometry for Physicits, Second Edition, World Scientific, 1999. ISBN: 9810235623. [ Electronic Version ]
Jerrold Marsden, Lectures on Mechanics, Cambridge University Press, 1992. ISBN: 0521428440. [ Draft of Second Edition ]
Jerrold Marsden, Tudor Ratiu, Introduction to Mechanics and Symmetry, Second Edition, Springer-Verlag, 2002. ISBN: 038798643X. [ Electronic Version ]

Grading

Your grade in the course is based on your project and your 20-minute project presentation during the time of the final.
The topic of your project should be decided in consultation with the instructor before the end of the first month of class.