Invariant tori of Hamiltonian systems that are nondegenerate in Rüssmann's sense (Q1374999)

In this paper a simple proof of the following theorem on the existence of invariant tori in perturbed Hamiltonian systems, originally announced by \textit{H. Rüssmann} [Number Theory and Dynamical Systems, Lond. Math. Soc. Lect. Note Ser. 134, 5-18 (1989; Zbl 0689.34039)], is presented: Consider the Hamiltonian \[ H(p,q,\varepsilon)=f(p)+ \varepsilon g(p,q,\varepsilon) \] where \(p \in S \subset {\mathbb R}^n\) and \(q \in T^n\) which is continuous with respect to \(\varepsilon \in {\mathbb R}\). If the image of the frequency map \(p \mapsto \omega(p)=\partial f/\partial p\) is not contained in any linear hyperplane passing through 0, then, for any perturbation \(\varepsilon g\), for all \(\varepsilon\) sufficiently small, \(H\) has invariant analytic \(n\)-dimensional tori close to the unperturbed tori \(p=\)const., which support quasiperiodic motions with incommensurable frequencies. The proof is based on a theorem of \textit{J. Pöschel} [Commun. Pure Appl. Math. 35, 653-696 (1982; Zbl 0542.58015)] and the theory of Diophantine approximations on submanifolds of the Euclidean space.