Collision and escape orbits in a generalized Hénon–Heiles model

The motion of a material point of unit mass in a field determined by a generalized Hénon–Heiles potential $U=A{q}_1^2+B{q}_2^2+C{q}_1^2{q}_2+D{q}_2^3$, with $(q_1,q_2)=$ standard Cartesian coordinates and $(A,B,C,D)\in {(0,\infty )}^2\times R^2$, is addressed for two limit situations: collision and escape. Using McGehee-type transformations, the corresponding collision and infinity boundary manifolds pasted on the phase space are determined. These are fictitious manifolds, but, due to the continuity with respect to initial data, their flow determines the near by orbit behaviour.The dynamics on the collision and infinity manifolds is fully described. The topology of the flow on the collision manifold is independent of the parameters. In the full phase space, while spiraling collision orbits are present, most of the orbits avoid collision. The topology of the flow on the infinity manifold changes as the ratio between $C$ and $D$ varies. More precisely, there are two symmetric pitchfork bifurcations along the line $2C?3D=0$, due to the reshaping of the potential along the bifurcation line. Besides rectilinear and spiraling orbits, the near-escape dynamics includes oscillatory orbits, for which angular momentum alternates sign.     

EDXRF spectrometry determination of tungsten in tobacco plants after antiviral treatment with 12-tungstophosphoric acid and its compounds

In this study, we have developed a sensitive, rapid and simple procedure for the energy dispersive X-ray fluorescence (EDXRF) spectrometry measurement of tungsten in tobacco plant parts. Only 0.1 g of dried plant material is needed instead of the usual 1 g. EDXRF spectrometry is used for quantitative measurement, after the foliar application of solutions of tungstophosphoric acid (WPA), its magnesium salt and compounds with glycine (Gly) and alanine (Ala), in exact quantities. After that, the leaves, trunks and summits were collected and prepared separately. Tungsten is determined directly in raw dried material, and the overlap of the tungsten peak with zinc's that is present is avoided by the spectral deconvolution to obtain quantitative results. The prepared dry tablets weighed 100 mg, and measurement time was 2000 s. The radioisotope excitation source used was ¹⁰⁹Cd and tungsten was identified and quantified at the L_α1 and L_α2 lines at the energies of 8397.6 eV and 8335.2 eV, respectively. EDXRF spectrometry was applied in a wide range of concentrations (up to 2000 mg/kg), with an estimated detection concentration limit of 15 mg/kg, calculated on dried material. Quantitative analysis of different parts of the treated plant plus the washings gave 94.47% recovery of the applied tungsten in different compound forms. After the foliar application of investigated WPA compounds, there were noticed both vertical and horizontal distributions of tungsten through out the tobacco plants, according to the EDXRF spectrometry results. This conclusion is also in agreement with the positive effects of WPA on Tobacco mosaic tobamovirus (TMV) infection of Nicotiana tabacum (Solanaceae).     

The two-body problem with generalized Lennard-Jones potential

We consider a generalization of the famous Lennard-Jones potential. To study the two-body problem associated to this potential, we use the foliations of the phase space by the invariant sets corresponding to the first integrals of energy and angular momentum. We investigate all possible situations created by the interplay among the constants of integration and the field parameters. We obtain the global flow, and illustrate it in both 3D and 2D. This flow exhibits a great variety of orbits, a homoclinic one included. All phase portraits are interpreted in terms of physical trajectories.