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We present an algorithm for the construction of lower dimensional elliptic tori in parametric Hamiltonian systems by means of the parametrization method with the tangent and normal frequencies being prescribed. This requires that the Hamiltonian system has as many parameters as the dimension of the normal dynamics, and the algorithm must adjust these parameters. We illustrate the methodology with an implementation of the algorithm computing $2$--dimensional elliptic tori in a system of $4$ coupled pendula (4 degrees of freedom).

We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement to the normal form. Using this result, we prove the Herman invariant tori conjecture namely the existence of a positive measure set of invariant tori near the critical point. This paper is an update of the first 2012 proof of the author. The functional analytic arguments have been simplified using Banach functors, minor points have been clarified. A series of videos is available on the webpage https://www.agtz.mathematik.uni-mainz.de/category/alg-geom/

This is part I of a book on KAM theory. We start from basic symplectic geometry, review Darboux-Weinstein theorems action angle coordinates and their global obstructions. Then we explain the content of Kolmogorov's invariant torus theorem and make it more general allowing discussion of arbitrary invariant Lagrangian varieties over general Poisson algebras. We include it into the general problem of normal forms and group actions. We explain the iteration method used by Kolmogorov by giving a finite dimensional analog. Part I explains in which context we apply the theory of Kolmogorov spaces which will form the core of Part II., Comment: This text is an extended version of ArXiv 1506.02514 part I

A Gevrey symplectic normal form of an analytic and more generally Gevrey smooth Hamiltonian near a Lagrangian invariant torus with a Diophantine vector of rotation is obtained. The normal form implies effective stability of the quasi-periodic motion near the torus.

We prove an analytic KAM-Theorem, which is used in [1], where the differential part of KAM-theory is discussed. Related theorems on analytic KAM-theory exist in the literature (e. g., among many others, [7], [8], [13]). The aim of the theorem presented here is to provide exactly the estimates needed in [1]., Comment: 48 pages

Chow, Li and Yi in [2] proved that the majority of the unperturbed tori {\it on sub-manifolds} will persist for standard Hamiltonian systems. Motivated by their work, in this paper, we study the persistence and tangent frequencies preservation of lower dimensional invariant tori on smooth sub-manifolds for real analytic, nearly integrable Hamiltonian systems. The surviving tori might be elliptic, hyperbolic, or of mixed type., Comment: 25 pages. To appear in Nonlinear Analysis: TMA

We propose a model for local dynamics of a perturbed convex real-analytic Liouville-integrable Hamiltonian system near a resonance of multiplicity $1+m, m\geq 0$. Physically, the model represents a toroidal pendulum, coupled with a Liouville-integrable system of $n$ non-linear rotators via a small analytic potential. The global bifurcation problem is set-up for the $n$-dimensional isotropic manifold, corresponding to a specific homoclinic orbit of the toroidal pendulum. The splitting of this manifold can be described by a scalar function on an $n$-torus, whose $k$th Fourier coefficient satisfies the estimate $$O(e^{- \rho|k\cdot\omega| - |k|\sigma}), k\in\Z^n\setminus\{0\},$$ where $\omega\in\R^n$ is a Diophantine rotation vector of the system of rotators; $\rho\in(0,{\pi\over2})$ and $\sigma>0$ are the analyticity parameters built into the model. The estimate, under suitable assumptions would generalize to a general multiple resonance normal form of a convex analytic Liouville integrable Hamiltonian system, perturbed by $O(\eps)$, in which case $\omega_j\sim\omeps, j=1,...,n.$, Comment: 24 pages

We show that near a focus-focus point in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy-momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy-momentum map that is transversal to the line of constant energy. In contrast to this we also show that the frequency map is non-degenerate for every point in a neighborhood of a focus-focus point., Comment: 13 pages

Written with respect to an appropriate Poisson structure, a partially integrable Hamiltonian system is viewed as a completely integrable system with parameters. Then, the theorem on quasi-periodic stability in Ref. [1] (the KAM theorem) can be applied to this system., Comment: 4 pages