Advanced Calculus fi..

Chapter 10 Partial Differential Equations 663the particles to be subject to resistances -hl(dxl/dt), -h2(dx2/dt) and to outsideforces Fl(t), F2(t). The differential equations have fhr: formThey simplify to the following:We have not explicitly stated that the walls are fixed at xo, x3, and the differentialequations remain correct even if the walls are moved in a manner controlled fromoutside the system. However, if .xu and x3 are constants x,*, x; and Fl(t), F2(t) are 0,the system has precisely one equilibrium state, namely, the solution of the equationsThe solution is at once found to beat equilibrium the particles are equally spaced between the walls.We now refer each particle to its equilibrium position by introducing new variables:as suggested in Fig. 10.2. The differential equations (10.21) then have the appearanceWhen the walls are fixed, xo = xg*, x3 = x;, uo and u3 are 0; however, (10.25) allowsfor a general motion of the walls and, in particular, for constant nonzero values ofuo and 142, which would signify a shift of the equilibrium position of the walls.We now consider several special cases of (10.25), paralleling the cases (a), (c),(d), (e), and (f) of the preceding section. The discussion of the analogue of (b) is leftto Problem 9. f ofFigure 10.2 Displacements from equilibruun positions, , , ,. r * A 481 i