## MAT983 Junior Seminar on Hamiltonian Mechanics, Fall-2013

### Announcements

Final paper is due on Dean's date

### Schedule

Each meeting is 2.30 hours, with three talks of 45mn each.
Thus at the nth meeting there will be talk number 2n-1 followed by talk number 2n. (see further below for the list of talks)
1. Sep-30 Mo
2. Oct-4 Fr
3. Oct-7 Mo
4. Oct-11 Fr
5. BREAK
6. Nov-4 Mo
7. Nov-8 Fr
8. Nov-15 Fr
9. Nov-22 Fr = office hours: your final paper
10. THANKSGIVING
11. Dec-6 Fr
Each student will deliver 2 talks of 45mn during the semester, for example one before the break and one after the break.

### Preparing your talk and your final paper

Please read the syllabus for some recommendations on preparing your talk and writing your final paper.
Usually it seems a good idea to prepare one detailed proof for each talk. Presenting a proof makes a talk more convincing. There is little time to do more than one proof.
Do not hesitate to search the literature for extra references. And do a fine selection of the material you are going to present.

### An inspiring quote

Mathematical physics is the discipline of people who try to reach a deep understanding of physical phenomena by following the rigorous style and method of mathematics. It is a discipline that lies at the border between physics and mathematics. The purpose of mathematical physicists is not to calculate phenomena quantitatively but to understand them qualitatively. They work with theorems and proofs not with numbers and computers. Their aim is to qualify with mathematical precision the concepts upon which physical theories are built.

Freeman Dyson in From Eros to Gaia.

### List of talks & topics

1. Solving the Kepler problem (Thomas) sharing
• Chapter 2 of [Mathuna]
• Chapter 2 of [A]
• wikipedia
2. Area preserving Flows (James) sharing
• Cauchy's existence theorem
• Section 1.1 of [MZ]
• Sections 1.0 to 1.4 of [GH]
• Poincare's recurrence theorem: section 1.4 of [HZ] and section 16 of [A]
3. Lagrangian formulation (Wesley) sharing
• Section 1 of [S]
• Chapter 3 of [A]
4. n-body problem, integrals of motions (Isabelle) sharing
• Section 5 of [S]
5. Symplectic matrices (Jager) sharing
• section 1.1 of [HZ]
6. restricted body problems, integral of motions (Mateo) sharing
• Siegel book [S] Hill problem
• Mathuna [M] Euler and Vinti problems
• online search on restricted body problems
7. Hamiltonian formulation (Brian Tu) sharing
• Section 3 of [S]
• Section 1.2 of [MZ]
• Section 15 of [A]
8. Equivalence of the formulations, Legendre transformation (Christoph) sharing
• Sections 1.2 and 1.4 of [MZ]
• Section 14 of [A]
9. ---------FALL BREAK----------
10. Liouville form, angle-action coordinates (?)
• sections 49 and 50 of [A]
• Appendix 7 of [A]
11. Sundman theorems, with proofs, part I (Derek and Danni)
• Chapters 6,8,9,10,11 of [S]
12. Lagrange points (Thomas)
• Chapter 14 of [S]
13. Sundman theorems, part II (Derek and Danni)
14. Triple collision, with proofs (Isabelle)
• Chapters 12,13 of [S]
15. Liouville theorem (Ray)
• section 16 of [A]
16. Analytic solutions, power series expansion, example of the Hill problem (Mateo)
• [S]
17. Poincare sections or Poincare-Birkhoff fixed point theorem (James)
• Appendix 9 of [A]
18. List of topics for the seminar paper (Nicolas)
19. Stability, KAM theory (last meeting, Derek, Danni, Christoph, Wesley, Brian, Ray)
• linearization: Chapter 5 of [A]
• Appendix 8 of [A]
• Birkoff normal form: section 1.7 and 1.8 of [HZ]
• section 1.8 of [HZ]
• [M]

### Resources & Notes

We will mainly read Siegel's famous book [S] "Lectures on Celestial Mechanics"
If the previous link doesn't work, here are direct links the pdf files: Some introductory material:
Excellent resource on stability:
• [M] Moser, Stable and Random Motions in Dynamical Systems -- see in the library call:QB351.M74 2001. The preface is also of historical interest.
Good references:
General discussion: Historical aspects:
Recent developments:
• [HZ] Hofer and Zehnder Symplectic Invariants and Hamiltonian Dynamics (this is very advanced! Our goal would be only: the introductory Chapter 1 and in the Chapter 6 try understand the modern formulation of Poincare's last theorem and Arnold conjecture)
Applications to astronomy:

### Past announcements

[]There seems to be 3 possibilities: Monday Wednedsday afternoon and Friday afternoon. Please visit this new link to tell which time potentially works for you. The increment is 15mn. You can add comments explaining your preferences or conflicts if needed.]

[DONE: Please visit this link and tell me when is a good time for you.]

First meeting is Monday Sep 23 in Fine 601 at 5pm (overview + organization where we will decide on the topics and speakers, finishing around 7pm)

### maintenance

We have a blackboard page for the seminar but I probably won't use it much. It is best to have everything in one place. This webpage will be our main communication tool.
time