School GIS2007

School on Geometry of Integrable Systems

Hanoi, 02-06 April 2007

Lecturers: Alain Albouy, Tudor Ratiu, Tadashi Tokieda, Alexei Tsygvintsev.

Each lecturer will give a 3-session minicourse. Each session will last about 1 to 1.5 hours.

Preliminary programme of the school

* =  colloquium talk at Hanoi Institute of Mathematics
* = starts at 11h00
** = starts at 14h30

Titles and abstracts of minicourses:

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Alain Albouy: Some classical integrable systems

Abstract: We will choose some classical integrable systems, as a ball with arbitrary internal repartition of mass, rolling on a table. We will deduce the equations of motion from d’Alembert principle, which is the standard way to treat together holonomic and non-holonomic systems. Then we will “reduce” these equations to “quadratures”, thus giving the main step toward integration. All the material of this lecture is completely standard, but we will try to present clearly some ideas which are not explained in modern textbooks. We will propose lecture notes and exercises.

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Tudor Ratiu: Geometric Methods of Integration

The goal of this survey is to present various geometric methods for finding first integrals. A special place is taken by completely integrable systems which will be
defined and their properties presented. The general notion of a momentum map with not necessarily numerical values will be introduced and reduction
briefly discussed. Then non-Abelian integrability is presented and a method of obtaining the full flow of an integrable system is presented. If time permits,
geometric phases will be discussed.

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Tadashi Tokieda: The art of modeling in mechanics

We examine important yet often neglected issues such as—What phase space we do use? How do we write constraints? How do we choose a Lagrangian or a Hamiltonian?—and surprising problems where different models lead to different predictions distinguishable by experiment, or lead even to mathematical
non-well-posedness. In particular, whether a mechanical problem is integrable or not may depend on how we model it. The intent of the course is to offer as many concrete, usable results as time allows, with solved exercises and toy demonstrations.

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Alexei Tsgvintsev: The polynomial first integrals, how to find them.

Abstract: The aim of this quite informal course is to introduce different methods and algorithms helping to find polynomial integrals of algebraic vector fields. We will concentrate mainly on quadratic homogeneous vector fields where the analysis can be done in a more simple and elegant way. A great number of concrete examples from mechanics and physics will be provided. The course will use only basic linear algebra and some elementary notions from the differential calculus.
Plan (3 hours):

1. Introduction to polynomial vector fields. First integrals. Dynamics.
2. Kowalevskaya exponents and the apriori estimates on degrees of first integrals.
3. Examples and applications.

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