Normal forms for perturbations of systems possessing a Diophantine invariant torus

TL;DR

This paper revisits Moser's 1967 normal form theorem for perturbations of systems with Diophantine invariant tori, offering a new proof and extending the results to dissipative systems with applications in Celestial Mechanics.

Contribution

It provides a new proof based on an abstract inverse function theorem and introduces generalized normal forms for dissipative systems, extending classical theorems.

Findings

Existence of a normal form for perturbed systems with Diophantine tori.

Development of translated-torus and twisted-torus theorems for dissipative systems.

Applications to Celestial Mechanics and generalizations of Herman and Rüssmann's theorems.

Abstract

In 1967 Moser proved the existence of a normal form for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. In this paper we present a proof of existence of this normal form based on an abstract inverse function theorem in analytic class. The given geometrization of the proof can be opportunely adapted accordingly to the specificity of systems under study. In this more conceptual frame, it becomes natural to show the existence of new remarkable normal forms, and provide several translated-torus theorems or twisted-torus theorems for systems issued from dissipative generalizations of Hamiltonian Mechanics, thus providing generalizations of celebrated theorems of Herman and R\"ussmann, with applications to Celestial Mechanics.