On the representation of maps by Lie transforms

The problem of representing a class of maps in a form suited for application of normal form methods is revisited. It is shown that using the methods of Lie series and of Lie transform a normal form algorithm is constructed in a straightforward manner. The examples of the Scrh\"oder--Siegel map and of the Chirikov standard map are included, with extension to arbitrary dimension

Methods of algebraic manipulation in perturbation theory

We give a short introduction to the methods of representing polynomial and trigonometric series that are often used in Celestial Mechanics. A few applications are also illustrated.Comment: 37 pages, 10 figure

An extensive adiabatic invariant for the Klein-Gordon model in the thermodynamic limit

We construct an extensive adiabatic invariant for a Klein-Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant $a$, the evolution of the adiabatic invariant is controlled up to times scaling as $\beta^{1/\sqrt{a}}$ for any large enough value of the inverse temperature $\beta$. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.Comment: 60 pages. Minor corrections with respect to the first version. To appear in Annales Henri Poincar\'

On the convergence of an algorithm constructing the normal form for lower dimensional elliptic tori in planetary systems

We give a constructive proof of the existence of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov's normalization algorithm to the case of planetary systems, for which elliptic tori may be used as replacements of elliptic keplerian orbits in Lagrange-Laplace theory. With this paper we support with rigorous convergence estimates the semi-analytical work in our previous article (2011), where an explicit calculation of an invariant torus for a planar model of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous works on the same subject we exploit the characteristic of Lie series giving a precise control of all terms generated by our algorithm. This allows us to slightly relax the non-resonance conditions on the frequencies.Comment: 45 page

Improved convergence estimates for the Schr\"oder-Siegel problem

We reconsider the Schr\"oder-Siegel problem of conjugating an analytic map in $\mathbb{C}$ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension $n>1$. Assuming a condition which is equivalent to Bruno's one on the eigenvalues $\lambda_1,\ldots,\lambda_n$ of the linear part we show that the convergence radius $\rho$ of the conjugating transformation satisfies $\ln \rho(\lambda )\geq -C\Gamma(\lambda)+C'$ with $\Gamma(\lambda)$ characterizing the eigenvalues $\lambda$, a constant $C'$ not depending on $\lambda$ and $C=1$. This improves the previous results for $n>1$, where the known proofs give $C=2$. We also recall that $C=1$ is known to be the optimal value for $n=1$.Comment: 21 page

ON A THEOREM OF LYAPOUNOV

Sunto. Si mostra che un sistema Hamiltoniano nell'intorno di un punto di equilibrio, sotto condizione che gli autovalori soddisfino delle condizioni di non-risonanza del tipo di Melnikov, ammette una forma normale che rende evidente l'esistenza di una varietà invariante (locale) a due dimensioni sulla quale si hanno soluzioni note. Nel caso di un autovalore puramente immaginario tali soluzioni formano una famiglia periodica a due parametri che costituisce la continuazione naturale di un modo normale. Questo secondo risultatoè stato dimostrato in precedenza da Lyapounov. In questo lavoro si completa quello di Lyapounov dimostrando la convergenza della trasformazione dell'Hamiltoniana a forma normale e rimuovendo le restrizione che gli autovalori siano puramente immaginari. Abstract. It is shown that a Hamiltonian system in the neighbourhood of an equilibrium may be given a special normal form in case the eigenvalues of the linearized system satisfy nonresonance conditions of Melnikov's type. The normal form possesses a two dimensional (local) invariant manifold on which the solutions are known. If the eigenvalue is pure imaginary then these solutions are the natural continuation of a normal mode of the linear system. The latter result was first proved by Lyapounov. The present paper completes Lyapounov's result in that the convergence of the transformation of the Hamiltonian to a normal form is proven and the condition that the eigenvalues be pure imaginary is removed

A KEPLER'S NOTE ON SECULAR INEQUALITIES

I discuss the problem of secular inequalities in Kepler by giving account of a manuscript note that has not been published until 1860. In his note Kepler points out the need for a model, clearly inspired by the method of epicycles, that describes the secular inequalities as periodic ones. I bring attention to this point, that seems to have been underestimated, since the references to Kepler's work usually report only that he observed a decreasing mean motion for Saturn and an increasing one for Jupiter

Normal Form and Energy Conservation of High Frequency Subsystems Without Nonresonance Conditions

We consider a system in which some high frequency harmonic oscillators are coupled with a slow system. We prove that up to very long times the energy of the high frequency system changes only by a small amount. The result we obtain is completely independent of the resonance relations among the frequencies of the fast system. More in detail, denote by $\epsilon^{-1}$ the smallest high frequency. In the first part of the paper we apply the main result of [BG93] to prove almost conservation of the energy of the high frequency system over times exponentially long with ${\epsilon^{-1/n}}$ ($n$ being the number of fast oscillators). In the second part of the paper we give e new self-contained proof of a similar result which however is valid only over times of order $\epsilon^{-N}$ with an arbitrary $N$. Such a second result is very similar to the main result of the paper [GHL13], which actually was the paper which stimulated our work