Mathematical Methods for Physics and Engineering : A Comprehensive Guide

9.4 EXERCISES

the figure and obtain three linear equations governing the currentsI 1 ,I 2 andI 3.

Show that the only possible frequencies of self-sustaining currents satisfy either

L L

C C

C I

1

I 2 I 3

P Q

S T R

U

(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, draw
an 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 formsuTAuanduTBu,whereuT
stands for the row matrix (xyz), and denote byunthose column matrices
that satisfy
Bun=λnAun, (E9.1)
in whichnis a label and theλnare real, non-zero and all different.
(a) By multiplying (E9.1) on the left by (um)T, and the transpose of the corre-
sponding equation forumon the right byun, show that (um)TAun=0for
n=m.
(b) By noting thatAun=(λn)−^1 Bun, deduce that (um)TBun=0form=n.
(c) It can be shown that theunare linearly independent; the next step is to
construct a matrixPwhose columns are the vectorsun.
(d) Make a change of variablesu=Pvsuch thatuTAubecomesvTCv,anduTBu
becomesvTDv. Show thatCandDare diagonal by showing thatcij=0if
i=j, and similarly fordij.
Thusu=Pvorv=P−^1 ureduces both quadratics to diagonal form.
To summarise, the method is as follows:
(a) find theλnthat allow (E9.1) a non-zero solution, by solving|B−λA|=0;
(b) for eachλnconstructun;
(c) construct the non-singular matrixPwhose columns are the vectorsun;
(d) make the change of variableu=Pv.

9.7 (It is recommended that the reader does not attempt this question until exercise 9.6
has been studied.)
If, in the pendulum system studied in section 9.1, the string is replaced by a
second rod identical to the first then the expressions for the kinetic energyTand
the potential energyVbecome(tosecondorderintheθi)
T≈Ml^2

( 8

3 ̇θ

2

1 +2 ̇θ^1 ̇θ^2 +

2

3 ̇θ

2

2

)

,

V≈Mgl

( 3

2 θ

2

1 +

1

2 θ

2

2

)

.

Determine the normal frequencies of the system and find new variablesξandη

that will reduce these two expressions to diagonal form, i.e. to

a 1 ̇ξ^2 +a 2 ̇η^2 and b 1 ξ^2 +b 2 η^2.