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A simple proof of the non-integrability of the first and the second Painlevé equations

Henryk Żołądek (2011)

Banach Center Publications

The first and the second Painlevé equations are explicitly Hamiltonian with time dependent Hamilton function. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems in ℂ⁴. We prove that the latter systems do not have any additional algebraic first integral. In the proof equations in variations with respect to a parameter are used.

Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials

Guillaume Duval, Andrzej J. Maciejewski (2009)

Annales de l’institut Fourier

In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential V ( q ) , q n , of degree k . The known results of Morales and Ramis give necessary conditions for the complete integrability of such systems. These conditions are expressed in terms of the eigenvalues of the Hessian matrix V ( c ) calculated at a non-zero point c n , such that V ( c ) = c . The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix V ( c ) is not diagonalizable....

Morales-Ramis Theorems via Malgrange pseudogroup

Guy Casale (2009)

Annales de l’institut Fourier

In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.

Non-integrability of certain Hamiltonian systems. Applications of the Morales-Ramis differential Galois extension of Ziglin theory

Andrzej J. Maciejewski (2002)

Banach Center Publications

The aim of this paper is to present two examples of non academic Hamiltonian systems for which the Morales-Ramis theory can be applied effectively. First, we investigate the Gross-Neveu system with n degrees of freedom. Till now it has been proved that this system is not integrable for n = 3. We give a simple proof that it is not completely integrable for an arbitrary n ≥ 3. Our second example is a natural generalisation of the Jacobi problem of a material point moving on an ellipsoid. We formulate...

On the classical non-integrability of the Hamiltonian system for hydrogen atoms in crossed electric and magnetic fields

Robert Gębarowski (2011)

Banach Center Publications

Hydrogen atoms placed in external fields serve as a paradigm of a strongly coupled multidimensional Hamiltonian system. This system has been already very extensively studied, using experimental measurements and a wealth of theoretical methods. In this work, we apply the Morales-Ramis theory of non-integrability of Hamiltonian systems to the case of the hydrogen atom in perpendicular (crossed) static electric and magnetic uniform fields.

Overview of the differential Galois integrability conditions for non-homogeneous potentials

Andrzej J. Maciejewski, Maria Przybylska (2011)

Banach Center Publications

We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form H = 1 / 2 i = 1 n p ² i + V ( q ) , where the potential V is a finite sum of homogeneous components. In this paper we show how to find, in the differential Galois framework, computable necessary conditions for the integrability of such systems. Our main result concerns potentials of the form V = V k + V K , where V k and V K are homogeneous functions of integer degrees k and K > k, respectively. We present examples of integrable...

Small divisors and large multipliers

Boele Braaksma, Laurent Stolovitch (2007)

Annales de l’institut Fourier

We study germs of singular holomorphic vector fields at the origin of n of which the linear part is 1 -resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms....

Summability of first integrals of a C ω -non-integrable resonant Hamiltonian system

Masafumi Yoshino (2012)

Banach Center Publications

This article studies the summability of first integrals of a C ω -non-integrable resonant Hamiltonian system. The first integrals are expressed in terms of formal exponential transseries and their Borel sums. Smooth Liouville integrability and a relation to the Birkhoff transformation are discussed from the point of view of the summability.

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