
Textbooks

Arnol'd V.I., (1989), Mathematical Methods of Classical Mechanics, SpringerVerlag,
Abraham R., Marsden J.E. (1978), Foundations of Mechanics, London: BenjaminCummings. Arnol'd V. I., Kozlov V. V., Neïshtadt A. I., (1988), Mathematical Aspects of Classical and Celestial Mechanics, 3, SpringerVerlag. Goldstein, H., (2002), Classical Mechanics, Addison Wesley. Landau L. D., Lifshitz L. M., (1975), Mechanics, Amsterdam, Elsevier. 
Office Hours

Office Hours: Monday 2:303:30pm Skiles 132. 


Report
(to main page ) 
Each student is requested to choose a topic related to the content of the course,
in agreement with the Instructor, for a 2025 pages written report. The content of this report will be explained by the student to the class in a 40 minutes presentation. This report will be submitted to the Instructor before the talk and NOT after April 29, 2013. This talk will be scheduled during the last three weeks of the semester. The evaluation of the written report and of the quality of the talk will provide the grade for the Final Exam. To prepare the report: 1)Summary: a document of 1015 lines maximum, in a pdf format, including (i) a title, (ii) the author's name, (iii) one or two references (book or article) (iv) a short description. The summary must be sent to the Instructor by email before Friday February 15, 2013 at 12:05pm. 2)Preliminary Report: it is a document of more than 20 and less than 25 pages, presented in pdf format and sent to the Instructor. This preliminary version should be sent before Friday April 5, 12:05pm for corrections and comments. It should be organized as follows (i) A title and an abstract (5 lines maximum) (ii) An Introduction (2 pages maximum) explaining the problem, some history with references, (iii) Various Sections introducing the notations, the framework and (iv) At least one Section describing in detail the derivation of one result (either a proof, or a calculation, or the description of an experiment and its results) (v) A conclusion, summarizing the results and indicating some prospects. (vi) A list of references. A special attention will be given by the grader to evaluate whether the references have been read and correctly cited within the text. The instructor will read this preliminary version during the month of April and will send suggestions comments and corrections back to the student. 3)Final Report: to be sent before Monday April 29, 2013 at 6:00pm for final evaluation. Oral Presentation: The oral presentation will last 35 minutes and it is recommended to use a computer, either a Power Point presentation or a pdf one. 15 minutes will be used for questions and comments. The final grade will include an evaluation of the Talk, of the Summary and of the Final Report. 
Final Exam

April 29th 2013
The individual report written and presented by each student will be the basis for the final grade. No formal final exam will be offered. 
Completely Integrable Systems  5 weeks 
 Historical background
Kepler's Laws, Newton's Principles, gravitation law, planetary motion, first integrals.  Coupled Harmonic Oscillators ActionAngle variables, vibrational mode distribution.  Conserved Quantities and Symmetries Liouvill'es integrability, Liouville's Theorem, invariant tori. Angular Momentum and rotation invariance, Poisson's brackets, commutation rules, Noether's Theorem.  Inverse Spectral Method Lax pairs, the Toda lattice, KdV equations, a list of models 
Week 1
nnnn Week 2 ~ Week 3 ~ Week 4 ~ Week 5 ~nnnn 
Differential Geometry & Mechanics  4 weeks 
 Manifolds
Charts, change of charts, tangent and cotangent space, Phase space, systems with constraints, geometry of speed and momentum.  Fiber Bundles Vector bundles, sections, connections, horizontal transport, curvature, torsion. Examples: rolling ball, spinor field, the Hopf fibration, Bloch waves.  Differential Calculus Differential of functions, 1forms, 2forms, nforms, exterior calculus, Symplectic forms, Hamilton equations, canonical transformations, Liouville form. Integration of forms, Stokes Theorem, Poincaré invariants, homology, cohomology. Berry's phase, Chern numbers, Chern classes, integrality.  Group Action Lie groups, Lie algebra, commutation rules. Hamiltonian symmetries, moment map. Coadjoint representation as an example of completely integrable system. 
Week 6
nnnn Week 7 ~ nnnn Week 8 ~ nnnn nnnn nnnn Week 9 ~nnnn nnnn nnnn 
Global Properties  3 weeks 
 Long Time Behavior
The Poincaré recurrence theorem, concept of ergodicity, the Birkhoff Theorem. Time and ensemble averaging, foundation of Statistical Mechanics.  Phase Space Portrait Poincaré section, first return map, computer simulation, orbits. Periodic orbits, linear stability, Lyapounov exponents, hyperbolicity. Bifurcation, the Hopf bifurcation, the RuelleTakens Theorem.  Invariant Tori and Cantori Perturbation of invariant tori, the KAM theorem, the Poincaré conjecture, AubryMather's theory, the FrenkelKontorova model, Peierls barrier. 
Week 10
nnnn /font> Week 11 ~ nnnn nnn Week 12 ~ nnnn 
Student Report Presentation  3 weeks 
Each student will be asked tho prepare a report on a topic related with
the content of the course and will present his work in front of other in a 35 minute talk and 15 minutes for questions and comments. 