Hamiltonian Dynamics 
R. de la Llave 
Fall 96 

Brief description:

Hamiltonian mechanics is the mathematical description of many physical
phenomena. It lies at the root of many interesting mathematical
developments including differential equations, symplectic geometry and
calculus of variations.

We plan to introduce several different topics that are of current
interest in research. The main emphasis will be in perturbation and
variational methods though some other topics may be considered if there
is interest from students.

There will be assignments that will be either computational or
theoretical, depending on the taste of the students.

Review of differential geometry:
Differential forms and Cartan calulus. Basic facts about symplectic geometry.

Review of numerical methods:
Integrators of ode's. Solvers of equations. Minimization routines.
Graphing and printing from programs. Packages for dynamical systems.

Mathematical formulations of mechanics:
Newtonian formulation. Lagrangian formulation. Hamiltonian formulation.
The origins of Hamiltonian formalism. Optics.

Examples:
Kepler problem. 3 and N body problems. Spherical pendulum. Oscillators.
Integrable systems (Toda Lattice, Calogero system). Numerical
algorithms for eigenvalues as hamiltonian systems. Geodesic flows.
Hydrodynamics as a hamiltonian system (formally)

Transformation theory:
Normal forms. Generatic functions. Lie series. Deformation theory.

Perturbation theory:
Perturbation theory for periodic orbits. Lindsted series for
quasiperiodic orbits. Poincar'e's proof of the stable manifold theorem.
Canonical perturbation theory. Numerical implementation of Lindstedt
series. Numerical calculation of invariant manifolds with Poincar'e
method.

KAM theorem on persistence of smooth tori:
The Kolmogorov proof. The Arnol'd proof. (Presumably only a sketch)

Nehoroshev theorem and adiabatic invariants. 

Symmetries and reduction. 

Variational theory:
Lagrangian submanifolds and their intersections. Variational theory for
periodic orbits. Variational methods for quasi-periodic orbits. Morse
theory of geodesics.

Poincar'e-Birkhoff fixed point theorem:
Birkhoff version. Conley-Zehnder version. Franks version. Le Calvez
version. Poincar'e version. 

Other topics to be discussed if there
is time or to be assigned projects. Some possibilities are:

Aubry-Mather theory. Non-existence of invariant tori. Renormalization
description of breakdown of tori. Renormalization description of period
doubling. Systems of coupled oscillators. Infinite dimensional
hamiltonians.

Textbooks:
W. Thirring: A course in Mathematical Physics Vol I
V. I. Arnol'd Mathematical Methods of Classical Mechanics
V. I. Arnol'd Geometric methods in the theory of differential equations
G. Gallavotti The elements of mechanics.

The 9 volumes of the Encyclopaedia of Mathematical Sciences devoted to
dynamical systems also contain a wealth of modern material. They are
designed to be read in almost any order and contain lots of pointers
to modern literature. R. Abraham and Marsden Foundations of Mechanics
is a good book but somewhat uneven and it requieres careful selection
to avoid getting confused. J. Marsden Lectures on mechanics. Contains
a very nice selection of material related to reduction.

Consent of instructor required: NO