MAT983 Junior Seminar on Hamiltonian Mechanics, Fall-2013
The seminar meets on Fridays, 2.30-5pm in Fine Hall 601
Final paper is due on Dean's date
Each meeting is 2.30 hours, with three talks of 45mn each.
Each student will deliver 2 talks of 45mn during the semester, for example one before the break and one after the break.
- Sep-30 Mo
- Oct-4 Fr
- Oct-7 Mo
- Oct-11 Fr
- Nov-4 Mo
- Nov-8 Fr
- Nov-15 Fr
- Nov-22 Fr = office hours: your final paper
- Dec-6 Fr
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
- Solving the Kepler problem (Thomas)
- Chapter 2 of [Mathuna]
- Chapter 2 of [A]
- Area preserving Flows (James)
- 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]
- Lagrangian formulation (Wesley)
- Section 1 of [S]
- Chapter 3 of [A]
- n-body problem, integrals of motions (Isabelle)
- Symplectic matrices (Jager)
- restricted body problems, integral of motions (Mateo)
- Siegel book [S] Hill problem
- Mathuna [M] Euler and Vinti problems
- online search on restricted body problems
- Hamiltonian formulation (Brian Tu)
- Section 3 of [S]
- Section 1.2 of [MZ]
- Section 15 of [A]
- Equivalence of the formulations, Legendre transformation (Christoph)
- Sections 1.2 and 1.4 of [MZ]
- Section 14 of [A]
- Liouville form, angle-action coordinates (?)
- sections 49 and 50 of [A]
- Appendix 7 of [A]
- Sundman theorems, with proofs, part I (Derek and Danni)
- Chapters 6,8,9,10,11 of [S]
- Lagrange points (Thomas)
- Sundman theorems, part II (Derek and Danni)
- Triple collision, with proofs (Isabelle)
Liouville theorem (Ray)
- Analytic solutions, power series expansion, example of the Hill problem (Mateo)
- Poincare sections or Poincare-Birkhoff fixed point theorem (James)
- List of topics for the seminar paper (Nicolas)
- 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]
Resources & Notes
We will mainly read Siegel's famous book [S]
"Lectures on Celestial Mechanics"
Some introductory material:
Excellent resource on stability:
- [A] Arnold book, Mathematical Methods of Classical Mechanics
- [MZ] Moser and Zehnder, Notes on dynamical systems
- [GH] Guckenheimer and Holmes, Nonlinear oscillations
- Introduction to the n-body problem
- [M] Moser, Stable and Random Motions in Dynamical Systems -- see in the library call:QB351.M74 2001. The preface is also of historical interest.
- Arnold original papers: [A:proof] Proof of a theorem of Kolmogorov on the invariance of quasi-periodic motions,
and [A:small] Small denominators and the problem of stability
- Celletti, Stability and chaos in celestial mechanics
- Morbidelli, Modern celestial mechanics
Applications to astronomy:
- [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)
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)
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.