Example sheets:

- Ex1: pdf  (Cauchy problems and qualitative study of dynamical systems)

- Ex2: pdf  (Qualitative study of dynamical systems and canonical formalism for Ham. syst.)

- Ex3: pdf  (Central forces and Kepler’s two-body problem)

- Ex4: pdf  (Integrable systems)

Hamiltonian dynamical systems

Part III - Lent 2012

Lecturer: Dr. Alfonso Sorrentino

Office:     E 0.13 (CMS)

Email:     A.Sorrentino[at]dpmms. cam. ac. uk

News:  - Example Sheet IV is on-line.              

              - Summer school in dynamical systems: Barcelona, 28th May - 1st June 2012 (JISD2012). Write to me if you are interested.

Main references: (see also the above syllabus)

[A]    -     V. I. Arnol’d, Mathematical methods of classical mechanics, Springer-Verlag, 1989.

[MZ] -     J. Moser and E. Zehnder, Notes on dynamical systems, Courant Lecture Notes, AMS 2005

[HSD]  -    M. Hirsch, S. Smale, R. Devaney, Differential equations, dynamical systems and an introduction to chaos, Elsevier, 2004.

Syllabus (Part III course announcement): pdf

Lecture diary

 Lecture 1 (Th. 19 Jan. 2012): General overview of the course: what is a dynamical system? whad does Hamiltonian mean? Why should we study these systems?
 Lecture 2 (S. 21 Jan. 2012): Introduction to the qualitative study of dynamial systems. Existence and uniqueness results for Cauchy problems.  Example of non-uniqueness. Continuous dependence on the initial condition (Gronwall inequality). See [HSD].
 Lecture 3 (T. 24 Jan. 2012): Some remarks on non-autonomous systems. Extending solutions: maximal solutions and global solutions. Fixed points of a vector field. Local equivalence of vector fields (far from fixed points). See [HSD] and [MZ].
 Lecture 4 (Th. 26 Jan. 2012): Fixed points of a vector field: stable, unstable, attractive and asymptotically stable fixed points. Stability and Instability properties via the linearized vector field. Definition of Limit set. Poincare-Bendixon theorem (only statement). See [HSD]
 Lecture 5 (S. 28 Jan. 2012): Definition of Hamiltonian system. Conservation of energy. Hamiltonian flows preserve the volume. Poincare’s return theorem and some applications. Arnold’problem: find the distribution of the first digit ot 2^n (Bernford’s law). See [MZ] and [A]. (Article on “Fact and Fiction in EU-Governmental Economica Data” - Link)
 Lecture 6  (T. 31 Jan. 2012): Cancelled (to be made up in the future)
 Lecture 7 (Th. 2 Feb. 2012): A variational definition of the Hamiltonian flow (Hamilton’s variational principle). Canonical transformations and Generalised canonical transformations. Symplectic matrices. See [MZ] and [A].
 Lecture 8-9   (T. 7 Feb. 2012): Generating functions. Extending a (spatial) transformation to a canonical one. Poisson bracket. See [MZ] & [A].
 Lecture 9-10 (Th. 9 Feb. 2012): Poisson bracket. Canonical transformations and Poisson bracket. When is possible to extend a general transformation to canonical one. Poisson-commuting functions. Example of group actions and symmetries: invariance under translation and rotation. See [MZ] and [A].
 Lecture 11-12 (T. 14 Feb. 2012): Group actions and Integrals of motions: Noether’s theorem. Moment maps. An example of SL(2) action and geodesic motion on a sphere. See [MZ] .
 Lecture 12-13 (Th. 16 Feb. 2012): Kepler’s problem and its “equivalence” to the geodesic motion on a sphere. See [MZ].
 Lecture 14-15 (T. 21 Feb. 2012): Basic results and definitions in symplectic geometry and Hamiltonian vector fields on symplectic manifolds. See [MZ] and [A]. See also A. Cannas da Silva, “Lectures on Symplectic geometry”
 Lecture 15-16 (Th. 23 Feb. 2012): Integrable systems. Liouville-Arnol’d theorem (I). See [MZ] and [A].
 Lecture 17-18 (T. 25 Feb. 2012): Liouville-Arnol’d theorem (II). See [MZ] and [A].
 Lecture 18-19 (Th. 1 Mar. 2012): Nearly-integrable system. Formal perturbation theory and the cohomological equation.
 Lecture 20-21 (T. 6 Mar. 2012): Kolmogorov’s theorem (KAM) I. Some references for the KAM theory.        
                -     A. N. Kolomogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function.
                      Dokl. Akad. Nauk SSSR (N.S.)  98: 527–530, 1954.
                -     V. I. Arnol’d, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small
                      perturbation of the Hamiltonian. Uspehi Mat. Nauk  18  no. 5 (113), 13–40, 1963.
                -     L. Chierchia, A. N. Kolmogorov's 1954 paper on nearly-integrable Hamiltonian systems. 
                       Regul. Chaotic Dyn.  13  no. 2, 130–139, 2008.
 Lecture 21-22 (Th. 8 Mar. 2012): Proof of Kolmogorov’s theorem (Main step + ideas on the inductive step)
 Lecture 23-24 (T. 13 Mar. 2012): Overview on current research in Hamiltonian dynamics.http://onlinelibrary.wiley.com/doi/10.1111/j.1468-0475.2011.00542.x/abstracthttp://www.math.ist.utl.pt/~acannas/Books/lsg.pdfshapeimage_1_link_0shapeimage_1_link_1

Lectures:                T - Th, 11:00 -12:30   (MR14)

Office hours:          By appointment

Example classes:    M.6/2, F.24/2 & F.9/3: 3-4 pm (MR4)