A criterion for asymptotic straightness of force fields (Q1956556)

Equation of motion arising from the classical scattering problem for potentials decreasing sufficiently fast at infinity is investigated. It is obvious to be interested in criteria which assure that the relevant potential provides the paths of particles moving to infinity approaching straight lines as asymptotes. In this paper a new original criterion is developed and verified. This criterion makes possible to decide whether the particular potential has the special property being called the asymptotic straightness. The comprehensive state of the art of the scattering problem for potentials decreasing to zero at infinity is included as a motivation of this study. Then a theoretical background and problem formulation is presented. Basic definitions of asymptotical straightness and asymptotical straightness in particular directions for time limiting to infinity are introduced. The main part of the paper is devoted to theorem of the asymptotic straightness and its proof on the level of ``necessary and satisfactory condition''. The principal idea is based on a generalized energy integral in Riemann meaning. In subsequent part illustrating examples are discussed. Some classes of asymptotical straight force fields are presented together with parameter intervals putting potentials into or outside asymptotic straight category. Later the potential \(V(x)= (1+x_1^2+x_2^2)^{-\alpha/2} x_2\); \(\alpha\in[3/2,2]\) is analysed. Finally the potential \(V(x)= \sin(x_1^2+x_2^2+x_1 xs_2)^{1/8} / (x_1^2+x_2^2+x_1 x_2)^{1/2}\); \(|x|\geq 1\) is investigated. An implication of the energy integral definition (Riemann/Lebesgue) is thoroughly discussed. The list of references is adequate. Although the paper is highly theoretical, the main theorem enables applications in theoretical physics of potential field using easy analytical basis only and without any numerical support.