On the nonintegrability of the generalized van der Waals Hamiltonian system

Abstract

Solid-liquid equilibrium of dipolar heteronuclear hard dumbbells in a generalized van der Waals theory: Application to methyl chloride Describing van der Waals Interaction in diatomic molecules with generalized gradient approximations: The role of the exchange functionalIn this paper we aim to prove that, except for the three known cases, the uniparametric family of Hamiltonian systems defined by the generalized van der Waals potential is nonintegrable in the Liouville-Arnold sense. The proof is based on the theorem of Morales and Ramis about nonintegrability by differential Galois theory.

Generalized van der Waals Hamiltonian: Periodic orbits andC1nonintegrability

“…By Theorem 3 only the solutions (11), (12) and (13) provide periodic solutions of the differential system (9) when the corresponding determinant is non-zero. We note that the solutions in (11) (respectively (12)) define a unique periodic orbit, both orbits are different but their projection into the plane (x, X) is a circle of radius √ h * and into the plane (y, Y ) also is a circle of radius √ h * .…”

“…For the values β 2 = 1/4, 1, 4 the Hamiltonian system is integrable, see Farrelly et al [5] and Ferrer et al [3,12].…”

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

“…A similar reasoning (carried out for a different value of the energy of another physical problem) has been used earlier in [11]. This leads to a simplification of the total system and allows to meet assumptions of the Morales-Ramis theorem with respect to the potential form.…”

“…Maciejewski and Przybylska have studied a class of all meromorphically integrable 2D Hamiltonian systems [10]. Also, Mondéjar and Ferrer [11] have discussed the non-integrability of the generalized van der Waals Hamiltonian system recovering the result of [8]. Sawicki and Kuś [16] have studied classical non-integrability of a quantum chaotic Hamiltonian system originating from atomic physics and quantum optics.…”

“…The idea of applying Morales-Ramis theory to study integrability and non-integrability of the system has been also inspired by yet another recent work, where a classical non-integrability for a model Hamiltonian, which takes its origins in atomic physics, has been proved [16]. Approach adopted for the case of crossed electric and magnetic fields will be similar to the method considered for the case of van der Waals Hamiltonian systems [11].…”