 95457 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 nondegenerate 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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