95-457 R. de la Llave

On necessary and sufficient conditions for uniform integrability of families of Hamiltonian systems. (83K, Plain TeX) Oct 13, 95

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Abstract. In ``Les methodes nouvelles de la m\'echanique c\'eleste'' Ch. V, specially \S 81, H. Poincar\'e discussed an obstruction to uniform integrability of families of Hamiltonians. (That is, the existence of changes of variables analytic in the parameter $\epsilon$ and in the variables that make the family of Hamiltonians a function of only action variables). We examine his proof and discover that, for non-degenerate systems, this condition is also sufficient for the integrability to first order in the parameter (That is, there exist analytical changes of variables, analytic in $\epsilon$ so that the family in these new variables depends only on the action variables up terms which are $o(\epsilon)$.) This leads to the existence of obstructions in higher order. We show that the vanishing of the obstructions to order $n$ is sufficient for the existence of analytic and symplectic changes of variables analytic in the parameter $\epsilon$ that reduce the system to integrable up to errors of order $\epsilon^{n+1}$. Moreover, we show that the vanishing of all the obstructions means that the system is uniformly integrable. This answers the question posed by Poincar\'e at the end of his chapter V. We note that these obstructions have a geometric meaning and they are cohomology obstructions computed on periodic orbits of the system.

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