This course is a natural continuation of 22M:142 (nonlinear dynamics with numerical methods). Though the course is mostly about the theory of ordinary differential equations (ODEs), we may consider this theory in conjunction with some theoretical and practical aspects of numerical methods for ODEs. If you have a high interest in a specific topic related to ODEs, please let me know. If a majority of students taking the class is interested in the topic then I may cover it during Spring semester 2014.

ASSIGNMENTS/HOMEWORK

1. hmwk1.ps, hmwk1.pdf due 09/06/13.
2. hmwk2.ps, hmwk2.pdf due 09/13/13.
3. hmwk3.ps, hmwk3.pdf due 09/20/13.
4. hmwk4.ps, hmwk4.pdf due 09/27/13.
5. hmwk5.ps, hmwk5.pdf due 10/04/13.
6. hmwk6.ps, hmwk6.pdf due 10/11/13.
7. hmwk7.ps, hmwk7.pdf due 10/18/13.
8. hmwk8.ps, hmwk8.pdf due 10/25/13.
9. hmwk9.ps, hmwk9.pdf due 11/01/13.
10. hmwk10.ps, hmwk10.pdf due 11/08/13.
11. hmwk11.ps, hmwk11.pdf due 11/15/13.
12. hmwk12.ps, hmwk12.pdf due 11/22/13.
13. hmwk13.ps, hmwk13.pdf due 12/11/13.

RECOMMENDED READINGS FROM THE TEXTBOOKS (AND FROM OTHER REFERENCES)

• In Sagan: 1.1, 1.2, 1.4, 1.5, 1.6, 1.7, p.40
• In Sagan: 2.1, 2.2, 2.3, 2.4, 2.6, 2.9, pp. 100-102, on the first variation
• In Sagan: 7.1, 7.2, 7.3, 7.5, 7.6, on the second variation
• In Sagan: 2.11, 7.7, 7.8, 7.9, pp. 420-421, on generalizations to n functions
• In Arnold: Part I (Chapters 1 and 2) on Newtonian mechanics (as light reading)
• In Arnold: 12 on variational Calculus
• In Sagan: A2.15 on Hamilton's principle
• In Arnold: 13, 17, 19, and 21 on Lagrangian mechanics and variational principles
• In Sagan: A2.16 and Arnold: 20 on Noether's theorem
• In Sagan: 2.12 and Arnold: 14 and 15 on Hamiltonian systems
• In Arnold: Chapter 7 (32-36) on differential forms
• In Arnold: 37, 38, and 44 (D from p.238-E) on symplectic structure and Hamiltonian systems. Warning: 'symplectic transformation' in Arnold are called 'canonical transformation' which may be a source of confusion since 'canonical transformations' in the literature are sometimes defined differently and in this case are (slightly) more general than symplectic transformations
• In Arnold: 16 on volume preservation and the Poincare recurrence Theorem

GENERAL INFORMATION

• Syllabus
• ICON
• Textbooks:
  • Introduction to the Calculus of Variations by Hans Sagan, 1969 edition, 449 pages, Paperback, Dover Publications, ISBN-10: 0486673669, ISBN-13: 978-0486673660, list price: $19.95. Library references: Physics Library QA315 .S23 1992 and Mathematical Sciences Library QA315 .S23. Table of contents. The book on amazon.com.
  • Mathematical Methods of Classical Mechanics by Vladimir I. Arnol'd, Translated by K. Vogtmann and A. Weinstein, Springer, Series: Graduate Texts in Mathematics, Vol. 60, 2nd edition 1989, Corr. 4th printing (December 1, 2010), 535 pages, Softcover ISBN-10: 1441930876, ISBN-13: 978-1441930873, list price: $74.95. Library references: Engineering Library QA805 .A6813 1989a, Physics Library QA805 .A6813 1989, and Sciences Library QA805 .A6813 1989b. The book on amazon.com.
  • Mathematical Aspects of Classical and Celestial Mechanics by Vladimir I. Arnol'd, Valery V. Kozlov, Anatoly I. Neishtadt, Translated by E. Khukhro, Springer, Series: Encyclopaedia of Mathematical Sciences, Vol. 3, Softcover reprint of hardcover 3rd ed. 2006 edition (November 19, 2010), 531 pages, ISBN-10: 3540282467, ISBN-13: 978-3540282464, list price: $179.00. Library references: Sciences Library QA805 .A67913 2006. The book on amazon.com. Electronic version.

COURSE TOPICS

• Variational calculus
• Lagrange's and Hamilton's mechanics
• Noether's theorem
• Divergence free ODEs (Poincare recurrence theorem)
• Reversible systems
• Perturbations
• Existence and uniqueness
• Lyapunov-stability
• Differential inequalities
• Normal forms
• Unfolding of vector fields
• Invariant (stable, unstable, center) manifolds
• Canonical transformations and generating functions
• KAM (Kolmogorov-Arnold-Moser) theory
• Poisson systems
• Variational/symplectic integrators
• Structural stability and bifurcations
• Averaging
• Homoclinic orbits
• Melnikov's method
• Boundary value problems (Sturm-Liouville systems)
• etc.

LINUX COMPUTER ACCOUNTS

• Linux computer accounts will be made available on the Linux network in MLH (computer lab rooms B5 and 301). You can also use the NoMachine NX Client Windows software to access your Linux computer account remotely. To access the Linux network (linux.divms.uiowa.edu) from off campus, you are required to establish a VPN connection before connecting (follow these instructions to install the software required to establish such connections).
• Selected Unix commands.
• Class directory: /group/class/m_213. Group you belong to: m_213.
• If you do not know your password go to 303A MLH.
• Accounts for students who are not majoring in Computer Science, Mathematics, or Statistics and Actuarial Science will be deleted after the term has ended! Note the expiration date at the top of your Unix Account Information form.
• The directory for this class will be removed before the beginning of the next term! If there is anything in this directory that you would like to save, you must archive it to your own home directory.
• Getting Started in the Division of Mathematical Sciences

MATLAB

• Free electronic textbook Numerical Computing with MATLAB by Cleve Moler himself!, the founder of Matlab.
• Matlab Help is available directly from the Help menu of the Matlab window.
• A Very Elementary MATLAB Tutorial, MATLAB Tutorials from Other Universities, MATLAB & Simulink Tutorials, MATLAB Homework Helper, all presented by Mathworks, Inc., producers of Matlab.
• Crash Course in MATLAB by Tobin Driscoll, University of Delaware.
• Some Matlab tutorials from Edward Neuman
• Matlab tutorial - University of New Hampshire
• Matlab tutorial - University of British Columbia
• Matlab tutorial - University of Florida
• MATLAB Online Reference Documentation provides direct hypertext links to specific MATLAB function descriptions (from the Math Dept, University of Florida).
• Matlab Primer (ps) (for an earlier version of Matlab)
• Matlab 5 introduction (ps, see also html)
• MATLAB Guide by D. J. Higham and N. J. Higham, SIAM, Philadelphia, 2005 (ENGINEERING QA297 .H5217 2005)
• Introduction to Engineering Programming: in C, MATLAB and JAVA by Mark A. Austin.
• An excellent book on Matlab: MATLAB: An Introduction with Applications by Amos Gilat, Hoboken, N.J.: Wiley ; Chichester : John Wiley, 2008, ISBN-10: 0470108770, ISBN-13: 978-0470108772 (ENGINEERING QA297 .G48 2008)

FORTRAN

• Type "man f90" on Unix workstations for information on how to compile
• A list of Fortran tutorials,

GRADER

• Rebecca Gasper , e-mail: rebecca-gasper@uiowa.edu, gasper.rebecca@gmail.com. 325N MLH. Phone: 335-3772.