Mainly drawing on explicit examples, the author introduces
the reader to themost recent techniques to study finite and
infinite dynamical systems. Without any knowledge of
differential geometry or lie groups theory the student can
follow in a series of case studies the most recent
developments. r-matrices for Calogero-Moser systems and Toda
lattices are derived. Lax pairs for nontrivial infinite
dimensionalsystems are constructed as limits of classical
matrix algebras. The reader will find explanations of the
approach to integrable field theories, to spectral transform
methods and to solitons. New methods are proposed, thus
helping students not only to understand established
techniques but also to interest them in modern research on
dynamical systems.

The Projection Method of Olshanetsky and Perelomov.- Classical Integrability of the Calogero-Moser Systems.- Solution of a Quantum Mechanical N-Body Problem.- Algebraic Approach to x 2 + ?/x 2 Interactions.- Some Hamiltonian Mechanics.- The Classical Non-Periodic Toda Lattice.- r-Matrices and Yang Baxter Equations.- Integrable Systems and gl(?).- Infinite Dimensional Toda Systems.- Integrable Field Theories from Poisson Algebras.- Generalized Garnier Systems and Membranes.- Differential Lax Operators.- First Order Differential Matrix Lax Operators and Drinfeld-Sokolov Reduction.- Zero Curvature Conditions on W ?, Trigonometrical and Universal Enveloping Algebras.- Spectral Transform and Solitons.- Higher Dimensional ?-Functions.