The Chemistry Maths Book, Second Edition

550 Chapter 19The matrix eigenvalue problem

then corresponds to a rotation of the coordinate system, from (x

1

,x

2

)to(y

1

,y

2

), to

bring the coordinate axes into coincidence with the principal axes of the ellipse, as

shown in Figure 19.2(b) forQ 1 = 19.

EXAMPLE 19.15The inertia tensor

A body rotating about an axis through its centre of mass with angular velocity xhas

kinetic energy of rotation given by the quadratic form

(19.43)

in which the symmetric matrixI(not to be confused with the unit matrix) is

called the moment of inertia tensoror, simply, the inertia tensor. For example,

a mass mat position (x, y, z) has the moment of inertia tensor with components

(see Exercise 35)

I

xx

1 = 1 m(y

2

1 + 1 z

2

), I

yy

1 = 1 m(z

2

1 + 1 x

2

), I

zz

1 = 1 m(x

2

1 + 1 y

2

)

I

xy

1 = 1 −mxy, I

yz

1 = 1 −myz, I

zx

1 = 1 −mzx

(19.44)

A principal-axis transformation is that coordinate transformation (x, y, z)to(x′, y′, z′)

that brings the coordinate axes into coincidence with the principal axes of inertia of

the body. The inertia tensor is diagonal in the new coordinate system, and the kinetic

energy of rotation is

(19.45)

and is the sum of contributions from rotations about thex′,y′,andz′axes.

Angular momentum

The angular momentum of a rotating body is related to the angular velocity by (see

Example 16.18)

l 1 = 1 Ix (19.46)

=++

′′ ′ ′′ ′ ′′ ′

1

2

1

2

1

2

222

III

xx x yy y zz z

ωωω

T

I

I

I

xyz

xx

yy

zz

=

′′′

′′

′′

′′















1

2

00

00

00

()ωωω























′

′

′









































ω

ω

ω

x

y

z

T

III

III

II

xyz

xx

xy xz

yx yy yz

zx zy

==

1

2

1

2

xx

T

I ()ωωω

II

zz

x

y

z



































































ω

ω

ω













