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MATH 277A Geometric Mechanics

MWF 3:00-3:50pm, APM 2402.

Instructor

Prof. Melvin Leok
Office: APM 5763
Email: mleok@math.ucsd.edu
Office Hours: MW 2:00pm-2:50pm, 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 one of the supplementary references, 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

Wednesday, June 10, 2015

3:00pm - 3:40pm	Ali Behzadan	Constrained Hamiltonian systems, gauge transformations, and general relativity
3:40pm - 4:00pm	Jeremy Schmitt	Hamilton-Jacobi theory and symplectic integrators
4:00pm - 4:20pm	Poorya Mirkhosrav	Multisymplectic discretization of the wave equation
4:30pm - 5:00pm	Daniel Copeland	The principle of gauge invariance in electrodynamics
5:00pm - 5:20pm	Pun Tong	Numerical calculation of the semi-classical limit of a Hamiltonian double-well system
5:20pm - 5:40pm	Yiqun Li	Variational integrators, Lie group methods and their applications
5:40pm - 6:00pm	Joseph Palmer	Variational principles for Lie-Poisson and Hamilton-Poincare equations
6:00pm - 6:30pm	Alexander Kuczala	The Hamilton-Jacobi-Bellman equation and dynamic programming

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

Jerrold Marsden, Tudor Ratiu, Introduction to Mechanics and Symmetry, Second Edition, Springer-Verlag, 2002. ISBN: 038798643X. [ 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.

Taeyoung Lee, Melvin Leok, Harris McClamroch, Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds, Springer-Verlag, under contract. [ Draft Version ]
Darryl Holm, Geometric Mechanics - Part I: Dynamics and Symmetry, Second Edition, Imperial College Press, 2011. ISBN: 184816775X. [ Electronic Version ]
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 ]

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.