Stationary point regular

FIGURE 8.4. Scattering curves for regular atoms and stationary point atoms. Point atoms are considered to have no width and no vibrational amplitude. Compare this Figure with Figure 3.13, Chapter 3. 

The solution of Underwood gave impetus to numerous investigations based on this approach. Part of these works was directed to the creation of geometric interpretation of the results obtained from the solution of the Underwood equations system. It is impossible without such interpretation to form a true notion of the general regularities of the distillation process of ideal mixtures. For one-section columns, geometric analysis of trajectories, stationary points, and separatrixes of distillation was realized even before the works of Underwood by Hausen (1934, 1935,1952) on the basis of calculations using the method tray by tray. ... 

In contrast to the product point, the pseudoproduct point can be located not only inside or at the boundary of the concentration simplex, but also outside it. The latter case refers to colunms of extractive distillation with two feeds, which leads to new regularities of location of trajectory bundles and their stationary points, that differ from regularities of location of top and bottom section trajectories. Therefore, we pay a lot of attention in this chapter to trajectory bundles of intermediate sections in extractive distillation colunms. 

The most severe topological objects of a PES are the stationary points minima and saddle points of any index between 1 and (n-1). Here, the gradient of E is zero, and this property overrides all regular coordinate transformations. 

We assume Newtonian mechanics to be valid and consider a mass point M bouncing in a two-dimensional square box of side length 1 (see Fig. 1.2). The box shown in Fig. 1.2 is used to illustrate regular motion. Therefore, we call it i . Another box is shown in Fig. 1.3. It is equipped with a hard stationary disk of radius r = 1/4 at its centre. It serves to illustrate chaotic motion. Therefore, we call it C7 . We assume that inside the boxes the mass point M travels on straight fine trajectories subject only to specular reflection whenever it hits the walls of R or C, or the central disk of box C. Since the motion of M is free between bounces, the mass of M is irrelevant for the kinematics of M. Therefore, M s velocity can be normalized to 1. 

Such schemes will typically look non-uniform and non-stationary when re-expressed in terms of points, but all the theory is still in fact regular when standing in the dual world. 

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