On a “mysterious” case of a quadratic Hamiltonian.
On the classical non-integrability of the Hamiltonian system for hydrogen atoms in crossed electric and magnetic fields
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.
On the concentration of solutions of singularly perturbed Hamiltonian systems in .
Overview of the differential Galois integrability conditions for non-homogeneous potentials
We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form , 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 , where and are homogeneous functions of integer degrees k and K > k, respectively. We present examples of integrable...