  This is part I of a book on KAM theory. We start from basic symplectic
geometry, review Darboux-Weinstein theorems action angle coordinates and their
global obstructions. Then we explain the content of Kolmogorov's invariant
torus theorem and make it more general allowing discussion of arbitrary
invariant Lagrangian varieties over general Poisson algebras. We include it
into the general problem of normal forms and group actions. We explain the
iteration method used by Kolmogorov by giving a finite dimensional analog.
  Part I explains in which context we apply the theory of Kolmogorov spaces
which will form the core of Part II.
