A C1 Arnol'd-Liouville theorem

Doklady Mathematics, 2006

2007

The Liouville-Arnold [22] theorem, stating that a Hamiltonian flow on a 2Ndimensional manifold with N different conservation laws in involution is integrable, is generalized to the following: Given an N-dimensional system of ordinary differential equations (ODE's) having (N − 1) Lie-Bäcklund symmetry generators, which together with the vector field given by the equation, do span at each point of the phase space an N-dimensional tangent space. Then this system is integrable (in the same sense as in the Liouville-Arnold theorem). Because of the fact that Hamiltonian systems have-in a natural way-a Noether operator [12], by which a map from gradients of conservation laws onto Lie-Bäcklund symmetry generators is given, the Liouville-Arnold theorem certainly is a special case of the result presented here (when applied to the level surfaces of the conservation laws).

Rendiconti Lincei - Matematica e Applicazioni, 2015

Consider a real-analytic nearly-integrable mechanical system with potential f , namely, a Hamiltonian system, having a real-analytic Hamiltonian H(y, x) = 1 2 |y| 2 + εf (x) , (1) y, x being n-dimensional standard action-angle variables (and | • | the Euclidean norm). Then, for "general" potentials f 's and ε small enough, the Liouville measure of the complementary of invariant tori is smaller than ε| ln ε| a (for a suitable a > 0).

E3S Web of Conferences

The paper studies the geometry of Liouville foliation generated by integrable Hamiltonian system. It is shown that regular leaves are two-dimensional surface of zero Gaussian curvature and zero Gaussian torsion.

Reports on Mathematical Physics, 2000

A symplectic theory approach is developed for solving the problem of algebraicanalytical construction of integral submanifold imbedding mapping for integrable via the abelian Liouville-Arnold theorem Hamiltonian systems on canonically symplectic phase spaces. The related Picard-Fuchs type equations are derived for the first time straightforwardly, making use of a method based on generalized Francoise-Galissot-Reeb differential-geometric results. The relationships between toruslike compact integral submanifolds of a LiouvilleArnold integrable Hamiltonian system and solutions to corresponding Picard-Fuchs type equations is stated. 1. General setting 1.1. Our main object of study will be differential systems of vector fields on the cotangent phase space M2" = T*(IW"), n E Z+, endowed with the canonical symplectic structure wc2) E A2(M2n), where ~(~1 = d(pr*c&')), and (Y(l) := (p, dq) = &dq,, j=1 (1.1) is the canonical l-form on the base space W", lifted naturally to the space A1 (Mzn), (q, p) E M2" are canonical coordinates on T*(B!"), pr : T*(IRY)-+ IR is the canonical projection, and (., .) is the usual scalar product in IIXn. Assume further that there is also given a Lie subgroup G (not necessarily compact), acting symplectically via the mapping cp : G x M2" + M2" on M2n, generating a Lie

Mathematische Zeitschrift, 1997

manuscripta mathematica, 2006

We show that a C 1 torus that is homologous to the zero section, invariant by the geodesic flow of a symmetric Finsler metric in T 2 , and possesses closed orbits is a graph of the canonical projection. This result, together with the result obtained by Bialy in 1989 for continuous invariant tori without closed orbits of symmetric Finsler metrics in T 2 , shows that the second Birkhoff Theorem holds for C 1 Lagrangian invariant tori of symmetric Finsler metrics in the two torus. We also study the first Birkhoff Theorem for continuous invariant tori of Finsler metrics in T 2 and give some sufficient conditions for a continuous minimizing torus with closed orbits to be a graph of the canonical projection.

Nonlinear Analysis-theory Methods & Applications, 2007

In this paper we formulate a theorem on the persistence of elliptic lower-dimensional invariant tori for nearly integrable analytic Hamiltonian systems under the first Melnikov condition and Rüssmann's nondegeneracy condition, and give the measure estimates of parameters for the non-resonance conditions under Rüssmann's non-degeneracy condition, which is essential for the proof of our result.

2016

Hamiltonians related to foliations, analogous to Riemannian foliations, are studied in the paper. One prove that each of the following data: a bundle–like Hamiltonian, a transverse hyperregular Hamiltonian, a hyperregular Hamiltonian foliated cocycle or a geodesic orthogonal property are equivalent to the fact that a foliation have to be a Riemannian one. Relations with the analogous Lagrangian case, considered previously by the authors, are studied.

arXiv (Cornell University), 2023

In this paper we present an a-posteriori KAM theorem for the existence of an (n − d)-parameters family of d-dimensional isotropic invariant tori with Diophantine frequency vector ω ∈ R d , of type (γ, τ), for n degrees of freedom Hamiltonian systems with (n−d) independent first integrals in involution. If the first integrals induce a Hamiltonian action of the (n − d)-dimensional torus, then we can produce n-dimensional Lagrangian tori with frequency vector of the form (ω, ωp), with ωp ∈ R n−d. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of size ρ, and the corresponding error in the functional equation is ε. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if γ −2 ρ −2τ −1 ε is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if γ −4 ρ −4τ ε is small enough. The approach is suitable to perform computer assisted proofs. Contents 1. Introduction 2. The setting and the KAM theorem 2.1. Basic notation 2.2. Symplectic setting 2.3. Hamiltonian systems and first integrals in involution 2.4. Invariant tori 2.5. Linearized dynamics and reducibility 2.6. A (modified) quasi-Newton method 2.7. Analytic setting 2.8. The KAM theorem 3. Proof of the KAM theorem 3.1. Some lemmas to control approximate geometric properties 3.2. One step of the iterative procedure 3.3. Convergence of the iterative process 4. Acknowledgements References Appendix A. An auxiliary lemma to control the inverse of a matrix Appendix B. Compendium of constants involved in the KAM theorem