Course announcement. 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 2010.

ASSIGNMENTS/HOMEWORK

1. hmwk1.ps, hmwk1.pdf due 09/09/09.
2. hmwk2.ps, hmwk2.pdf due 09/21/09.
3. hmwk3.ps, hmwk3.pdf due 10/05/09.
4. hmwk4.ps, hmwk4.pdf due 10/14/09.
5. hmwk5.ps, hmwk5.pdf due 10/30/09.
6. hmwk6.ps, hmwk6.pdf due 11/06/09.
7. hmwk7.ps, hmwk7.pdf due 11/13/09.
8. hmwk8.ps, hmwk8.pdf due 11/20/09.
9. hmwk9.ps, hmwk9.pdf due 12/09/09.

RECOMMENDED READINGS FROM THE TEXTBOOKS (AND FROM OTHER REFERENCES)

• In Gelfand: 1.1-1.4, 1.7 on variational Calculus and the first variation
• In Arnold: 12 on variational Calculus
• In Gelfand: 5.24-5.28 on the second variation
• In Gelfand: 2.9 and 5.29 on generalization to n unknown functions
• In Arnold: Part I (Chapters 1 and 2) on Newtonian mechanics (as light reading)
• In Arnold: 13, 17, 19, and 21 on Lagrangian mechanics and variational principles
• In Gelfand: 4.21 on Hamilton's principle
• In Arnold: 14 and 15 on Hamiltonian systems
• In Gelfand: 4.16-4.18 on the Hamiltonian formulation of Euler-Lagrange equations
• 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 defined differently and are (slightly) more general than symplectic transformations
• In Arnold: 20 on Noether's Theorem
• In Gelfand: 4.20 on Noether's Theorem and 4.22 on some first integrals
• In Arnold: 16 on volume preservation and the Poincare recurrence Theorem
• In Arnold: 44 on the Poincare-Cartan integral invariant

GENERAL INFORMATION

• Syllabus
• ICON
• Textbooks (recommended):
  • Calculus of Variations by I. M. Gelfand and S. V. Fomin, 2000, 240 pages, Paperback, Dover Publications, ISBN-10: 0486414485, ISBN-13: 978-0486414485, list price: $10.95. Library references: Mathematical Sciences Library QA315 .G417. The book on amazon.com.
  • Mathematical Methods of Classical Mechanics by V.I. Arnol'd, Springer, Series: Graduate Texts in Mathematics, Vol. 60, 2nd edition 1989, Corr. 4th printing (September 5, 1997), 509 pages, Hardcover, ISBN-10: 0387968903, ISBN-13: 978-0387968902, list price: $69.95. Library references: Engineering Library QA805 .A6813 1989a, Physics Library QA805 .A6813 1989, and Physics Reserve QA805 .A6813 1989b. Table of contents. The book on amazon.com.

COURSE TOPICS

• Existence and uniqueness (covered in 22M:213)
• Lyapunov-stability (covered in 22M:213)
• Variational calculus (covered in 22M:213)
• Lagrange's and Hamilton's mechanics (covered in 22M:213)
• Noether's theorem (covered in 22M:213)
• Divergence free ODEs (Poincare recurrence theorem) (covered in 22M:213)
• Reversible systems (covered in 22M:213)
• Perturbations (covered in 22M:214)
• Differential inequalities (covered in 22M:214)
• Normal forms (covered in 22M:214)
• Unfolding of vector fields (covered in 22M:214)
• Invariant (stable, unstable, center) manifolds (covered in 22M:214)
• Canonical transformations and generating functions (covered in 22M:214)
• KAM (Kolmogorov-Arnold-Moser) theory (covered in 22M:214)
• Poisson systems (covered in 22M:214)
• Variational/symplectic integrators
• Structural stability and bifurcations
• Averaging
• Homoclinic orbits
• Melnikov's method
• Boundary value problems (Sturm-Liouville systems)
• etc.

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.
• Matlab for first order ODEs
• Matlab for higher order ODEs and systems of ODEs
• Matlab for PDEs

FORTRAN

• Type "man f90" on Linux terminals for information on how to compile.
• A list of Fortran tutorials, Fortran 90 for the Fortran 77 programmer, Fortran 90 for Fortran 77 programmers.

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.
• 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. To change your Mathematical Sciences password here is the online password change tool.
• 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.

GRADER

• Beven Kair, e-mail: bkair@math.uiowa.edu. B20J MLH. Phone: 335-3650.