Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NORMAL MODES

P P P

l

l

θ 1

θ 1

θ 1

θ 2 θ^2

θ 2

(a) (b) (c)

Figure 9.1 A uniform rod of lengthlattached to the fixed pointPby a light

string of the same length: (a) the general coordinate system; (b) approximation

to the normal mode with lower frequency; (c) approximation to the mode with

higher frequency.

With these expressions forTandVwe now apply the conservation of energy,

d

dt

(T+V)=0, (9.6)

assuming that there are no external forces other than gravity. In matrix form

(9.6) becomes

d

dt

(q ̇TAq ̇+qTBq)=q ̈TA ̇q+ ̇qTA ̈q+ ̇qTBq+qTB ̇q=0,

which, usingA=ATandB=BT, gives

2 ̇qT(A ̈q+Bq)=0.

We will assume, although it is not clear that this gives the only possible solution,

that the above equation implies that the coefficient of each ̇qiis separately zero.

Hence

A ̈q+Bq= 0. (9.7)

For a rigorous derivation Lagrange’s equations should be used, as in chapter 22.

Nowwesearchforsetsofcoordinatesqthatalloscillate with the same period,

i.e. the total motion repeats itselfexactlyafter afiniteinterval. Solutions of this

form will satisfy

q=xcosωt; (9.8)

the relative values of the elements ofxin such a solution will indicate how each