Advanced Calculus fi..

Here we make a change of variable, setting y = Nx, so thatChapter 9 Ordinary Differential Equationswhere B = N-I AN-'. Since A is symmetric, we verify that B is also symmetric(Problem 2 below). Therefore B is similar to a diagonal matrix D = diag (A1, . . . , A,),where Al , . . . , An are the eigenvalues of B, which are also positive (Problem 2 below);we writeB=c-'DC,sothatD=C~C-'.If we now let y = C-'z, then we obtain z = Cy and- 19that isSince each A, is positive, we can write Ai = o;, and the solutions of (9.47) are givenbyzi=A,sin(wit+a;), i=l, ..., n, (9.48)where A,, . . . , A,, al , . . . , a, are arbitrary real constants. Now x = N-' y =N-'C-'z = Ez, for an appropriate matrix E = (el,). HencennThis is the general solution of (9.46). If, for example, A2 = . - . = A, = 0, thenx; = eilA1 sin(olt +al), i = 1 ,...,n, (9.50)and all coordinates x, oscillate synchronously with frequency ol. This is called anornzal mode of oscillation of the system. The general solution (9.49) can be regardedIas a superposition of normal modes.For physical examples of such vibrations, see Sections 10.3 and 10.4. ' ""Here A = [!: ;:I, and A has eigenvalues 16 and 4, so that V is positive definite andhas a minimum at (0,O). To find the solutions, we can simply set xi = u, sin (At +a,)in the differential equations, since we know that there are solutions of this form-thenormal modes. We obtain the equations(10 - 4h2)ul - 6u2 = 0, -6~1 + (10 - 4h2)u2 = 0.They are satisfied for h = f 1, h = f 2. For h = 1 we obtain ul = 1, u2 = 1;for A = 2 we obtain ul = 1, uz = - 1. Hence we obtain the normal modes.)