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

For an autonomous nearly integrable Hamiltonian system ofndegrees of freedom withn> 1 it was shown by Poincaré 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 (2n- 1)-dimensional manifolds exist, apart from constant-energy surfaces. On the other hand, the Kolmogorov-Amold-Moser theorem guarantees the existence ofn-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 ofntori with mutually proportional frequencies), in general such invariant manifolds do not exist.