An extension of the Poincaré-Fermi theorem on the nonexistence of invariant manifolds in nearly integrable Hamiltonian systems

Poincar\ue9 mostr\uf2 che per, un sistema hamiltoniano autonomo quasi integrabile adn gradi di libert\ue0, conn > 1, in generale non esistono integrali del moto indipendenti dalrhamiltoniana. Fermi generalizz\uf2 poi il risultato mostrando che, in generale, non esistono neppure singole variet\ue0 invarianti di dimensione 2n - 1, a parte le superfici di energia costante. D\u2019altro canto il teorema di Kolmogorov, Arnold e Moser garantisce l\u2019esistenza di tori invarianti di dimensionen. Nel presente lavoro si discute l\u2019eventuale esistenza di variet\ue0 invarianti di dimensione intermedia e si conclude che, a parte ben definite eccezioni (variet\ue0 cosiddette risonanti, e famiglie (n + l)-dimensionali din-tori, con frequenze mutuamente proporzionali), in g\ue9n\ue9rale tali variet\ue0 invarianti non esistono.For an autonomous nearly integrable Hamiltonian system of n degrees of freedom with n > 1 it was shown by Poincar\ue9 that, in general, no integrals of motion exist which are independent of the Hamiltonian. This result was generalized by Fermi, who showed that in general not even single invariant (2 n - 1)-dimensional manifolds exist, apart from constant-energy surfaces. On the other hand, the Kolmogorov-Amold-Moser theorem guarantees the existence of n-dimensional invariant tori. In this paper we discuss the possible existence of invariant manifolds of intermediate dimensions and conclude that, apart from very well-defined exceptions (namely, manifolds of the so-called resonant type and (n + 1)-dimensional families of n tori with mutually proportional frequencies), in general such invariant manifolds do not exist