- 99-105 Alex Haro
- The Primitive Function of an Exact Symplectomorphism.
Variational principles, converse KAM theory and the problems
of determination and interpolation.
(20554K, gzipped ps)
Apr 12, 99
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Abstract. This thesis has been made under the direction of Prof. Carles Simo.
The main contribution is the systematic use of the primitive
function of an exact symplectomorphism. The analytical, geometrical
and numerical tools used along this thesis take into account
the properties of this primitive function.
We have divided the thesis in four parts.
PART I: Exact symplectic geometry (introduction of the problems).
PART II: On the standard symplectic manifold (analytical part).
PART III: On the cotangent bundle (geometrical part).
PART IV: Applications (numerical part).
In last part we generalize converse KAM theory by MacKay,
Meiss and Stark and relate it with Lipschitz theory by Birkhoff,
Herman and Mather. We also perform a Greene method to detect the
breakdown of an invariant torus based upon variational principles.
We apply them to a broad class of examples: standard map, exponential
standard map, quadratic standard map, Froeschl\'e map (and its
family), rotational standard map (a symplectic skew-product), ...
We study numerically the Aubry-Mather sets in higher
dimensions. We also talk about
geometrical obstructions, normal forms, foliations, ...