By Kantorovich Aharon
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Extra info for An Ideal Model for the Growth of Knowledge in Research Programs
Draw two lines x r P , which intersect the x axis at points B and C respectively, as shown in Fig. 24. The phase paths consist of a series of circular arcs with the centers C or B, depending on whether the phase point is in the upper or the lower half-plane. In Fig. 24, a phase path is initiated at A. Draw a circle clockwise with center C and radius CA until it meets the x axis at A1. Then, switch the center to B and draw a half circle clockwise with a radius BA1 that meets the x axis again at A2.
1), we f ( x,0) 0 . 5) The constant solutions of the above algebraic equation are a set of equilibrium point coordinates of the system, which can be written as x0i, i 1, 2, " , Obviously, they are a discrete set on the real axis x. All of the equilibrium points and the phase paths in their neighborhood are the main components of the phase diagram of dynamic systems. 2 Phase Diagrams of Conservative Systems A conservative system is a special type of autonomous dynamic system and its phase diagram has some features.
Since both x(t) and 23 Chapter 2 Geometrical Method y(t) are periodic functions of time t, all these phase paths are closed curves. However, they are not harmonic functions. Thus, the phase paths are not ellipsoids. 9) where the force per unit mass f is independent of x . Define a function V(x), V ( x) ³ f ( x)dx or dV ( x) dx f ( x) . 10) Then V(x) is called potential energy function of the system. 9) are given by f ( x) V c( x) 0. 11) By writing x y in Eq. 3), and integrating the following equation, which is derived from Eq.