A Lagrangian proof of the invariant curve theorem for twist mappings (Q2778794)

This paper presents a Lagrangian formulation of the twist theorem for invariant curves, a result of permanent interest. The problem is to find invariant curves for certain diffeomorphims of the cylinder and it leads to the study of a system of two difference equations of first order. This was the original approach employed by Moser and we shall refer to it as the Hamiltonian approach.NEWLINENEWLINENEWLINEIn the present paper the diffeomorphism is exact symplectic and, if it satisfies a uniform twist condition, a global generating function can be found. With the help of this generating function, the search of invariant curves is now reduced to solve a second-order difference equation. The study of this equation is referred as the Lagrangian approach. The analogy with mechanics explains this terminology, which was also employed in a related context by \textit{D. Salamon} and \textit{E. Zehnder} in KAM theory in configuration space [Comment. Math. Helv. 64, No. 1, 84-132 (1989; Zbl 0682.58014)].NEWLINENEWLINENEWLINEThe Lagrangian formulation leads naturally to a notion of the approximate invariant curve. The main theorem of the paper assumes the existence of one of these approximate curves and concludes that, under appropriate conditions, there exists an invariant curve. The proof is obtained via KAM techniques. It is natural to compare this result with the standard twist theorem (in the analytic framework). The original version deals with diffeomorphisms having the intersection property. This class is more general than the class of exact-symplectic mappings. On the other hand, the present theorem is more flexible because the searched invariant curve is not necessarily close to a circle \(r=\) constant.NEWLINENEWLINENEWLINEThe paper also contains some discussions on the small twist theorem, the stability of elliptic fixed points and some related literature.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00044].