The geometry of nondegeneracy conditions in completely integrable systems

Details The geometry of nondegeneracy conditions in completely integrable systems The geometry of nondegeneracy conditions in completely integrable systems

2004-03-24

arxiv.org/abs/math/0403412v1

Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 4, 705--719.

Mathematics

Differential Geometry

Scientific paper

Nondegeneracy conditions need to be imposed in K.A.M. theorems to insure that the set of diophantine tori has a large measure. Although they are usually expressed in action coordinates, it is possible to give a geometrical formulation using the notion of regular completely integrable systems defined by a fibration of a symplectic manifold by lagrangian tori together with a Hamiltonian function constant on the fibers. In this paper, we give a geometrical definition of different nondegeneracy conditions, we show the implication relations that exist between them, and we show the uniqueness of the fibration for non-degenerate Hamiltonians.

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