Mathematical Methods for Physics and Engineering : A Comprehensive Guide

9.1 TYPICAL OSCILLATORY SYSTEMS

corresponding to a kinetic energy, is positive definite; that is, whatever non-zero

real values the ̇qitake, the quadratic form (9.1) has a value>0.

Turning now to the potential energy, we may write its value for a configuration

qby means of a Taylor expansion about the originq= 0 ,

V(q)=V( 0 )+

∑

i

∂V( 0 )

∂qi

qi+

1

2

∑

i

∑

j

∂^2 V( 0 )

∂qi∂qj

qiqj+···.

However, we have chosenV( 0 ) = 0 and, since the origin is an equilibrium point,

there is no force there and∂V( 0 )/∂qi= 0. Consequently, to second order in the

qiwe also have a quadratic form, but in the coordinates rather than in their time

derivatives:

V=

∑

i

∑

j

bijqiqj=qTBq, (9.2)

whereBis, or can be made, symmetric. In this case, and in general, the requirement

that the potential is a minimum means that the potential matrixB, like the kinetic

energy matrixA, is real and positive definite.

9.1 Typical oscillatory systems

We now introduce particular examples, although the results of this section are

general, given the above restrictions, and the reader will find it easy to apply the

results to many other instances.

Consider first a uniform rod of massMand lengthl, attached by a light string

also of lengthlto a fixed pointPand executing small oscillations in a vertical

plane. We choose as coordinates the anglesθ 1 andθ 2 shown, with exaggerated

magnitude, in figure 9.1. In terms of these coordinates the centre of gravity of the

rod has, tofirst orderin theθi, a velocity component in thex-direction equal to

l ̇θ 1 +^12 l ̇θ 2 andinthey-direction equal to zero. Adding in the rotational kinetic

energy of the rod about its centre of gravity we obtain, to second order in the ̇θi,

T≈^12 Ml^2 ( ̇θ^21 +^14 ̇θ 22 +θ ̇ 1 ̇θ 2 )+ 241 Ml^2 ̇θ^22

=^16 Ml^2

(

3 ̇θ^21 +3 ̇θ 1 ̇θ 2 +θ ̇^22

)

= 121 Ml^2 q ̇T

(

63

32

)

q ̇, (9.3)

where ̇qT=( ̇θ 1 ̇θ 2 ).The potential energy is given by

V=Mlg

[

(1−cosθ 1 )+^12 (1−cosθ 2 )

]

(9.4)

so that

V≈^14 Mlg(2θ^21 +θ^22 )= 121 MlgqT

(

60

03

)

q, (9.5)

wheregis the acceleration due to gravity andq=(θ 1 θ 2 )T; (9.5) is valid to

second order in theθi.