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Transactions of the American Mathematical Society

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Examples of nonintegrable analytic Hamiltonian vector fields with no small divisors

Author: R. Cushman
Journal: Trans. Amer. Math. Soc. 238 (1978), 45-55
MSC: Primary 58F05; Secondary 70.58
MathSciNet review: 0478223
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Abstract: Any analytic symplectic diffeomorphism $ \Phi $ of a symplectic manifold M is the Poincaré map of a real analytic Hamiltonian vector field $ {X_H}$. If $ \Phi $ does not have an analytic integral, then $ {X_H}$ has no analytic integral which is not a power series in H. Let $ M = {{\mathbf{R}}^2}$. If $ \Phi $ has a finite contact homoclinic point, then $ \Phi $ is nonintegrable. Also Moser's polynomial mapping is nonintegrable.

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Keywords: Symplectic diffeomorphism, suspension, integrable, Moser's diffeomorphism, homoclinic point
Article copyright: © Copyright 1978 American Mathematical Society