Mathematical Methods for Physics and Engineering - Matematica.NET

9.4 EXERCISESthe figure and obtain three linear equations governing the currents I 1 , I 2 and I 3 .Show that the only possible frequencies of self-sustaining currents satisfy eitherPCI 1QLULSI 2 TI 3CCR(a) ω 2 LC =1or(b)3ω 2 LC = 1. Find the corresponding current patterns and,in each case, by identifying parts of the circuit in which no current flows, drawan equivalent circuit that contains only one capacitor and one inductor.9.6 The simultaneous reduction to diagonal form of two real symmetric quadratic forms.Consider the two real symmetric quadratic forms u T Au and u T Bu, whereu Tstands for the row matrix (x y z), and denote by u n those column matricesthat satisfyBu n = λ n Au n ,(E9.1)in which n is a label and the λ n are real, non-zero and all different.(a) By multiplying (E9.1) on the left by (u m ) T , and the transpose of the correspondingequation for u m on the right by u n , show that (u m ) T Au n =0forn ≠ m.(b) By noting that Au n =(λ n ) −1 Bu n , deduce that (u m ) T Bu n =0form ≠ n.(c) It can be shown that the u n are linearly independent; the next step is toconstruct a matrix P whose columns are the vectors u n .(d) Make a change of variables u = Pv such that u T Au becomes v T Cv, andu T Bubecomes v T Dv. Show that C and D are diagonal by showing that c ij =0ifi ≠ j, and similarly for d ij .Thus u = Pv or v = P −1 u reduces both quadratics to diagonal form.To summarise, the method is as follows:(a) find the λ n that allow (E9.1) a non-zero solution, by solving |B − λA| =0;(b) for each λ n construct u n ;(c) construct the non-singular matrix P whose columns are the vectors u n ;(d) make the change of variable u = Pv.9.7 (It is recommended that the reader does not attempt this question until exercise 9.6has been studied.)If, in the pendulum system studied in section 9.1, the string is replaced by asecond rod identical to the first then the expressions for the kinetic energy T andthe potential energy V become(tosecondorderintheθ i )T ≈ Ml ( 2 8 ˙θ 2 3 1 +2˙θ 1˙θ2 + 2 ˙θ 2 3 2),V ≈ Mgl ( 32 θ2 1 + 1 2 2) θ2 .Determine the normal frequencies of the system and find new variables ξ and ηthat will reduce these two expressions to diagonal form, i.e. toa 1˙ξ2 + a 2˙η 2 and b 1 ξ 2 + b 2 η 2 .331