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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....

On the analytic non-integrability of the Rattleback problem

H. R. Dullin, A.V. Tsygvintsev (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

We establish the analytic non-integrability of the nonholonomic ellipsoidal rattleback model for a large class of parameter values. Our approach is based on the study of the monodromy group of the normal variational equations around a particular orbit. The imbedding of the equations of the heavy rigid body into the rattleback model is discussed.

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.

Viscosity solutions methods for converse KAM theory

Diogo A. Gomes, Adam Oberman (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

The main objective of this paper is to prove new necessary conditions to the existence of KAM tori. To do so, we develop a set of explicit a-priori estimates for smooth solutions of Hamilton-Jacobi equations, using a combination of methods from viscosity solutions, KAM and Aubry-Mather theories. These estimates are valid in any space dimension, and can be checked numerically to detect gaps between KAM tori and Aubry-Mather sets. We apply these results to detect non-integrable regions in several...

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