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In this paper, we give a new proof of the classical KAM theorem on thepersistence of an invariant quasi-periodic torus, whose frequency vectorsatisfies the Bruno-R\"ussmann condition, in real-analytic non-degenerateHamiltonian systems close to integrable. The proof, which uses rationalapproximations instead of small divisors estimates, is an adaptation to theHamiltonian setting of the method we introduced in a previous work forperturbations of constant vector fields on the torus.

The classical KAM theorem for Hamiltonian systems via rational approximations