Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.12 PHYSICAL APPLICATIONS OF TENSORS

section 26.7, sinceJandωare vectors). The tensor is called theinertia tensoratO

of the assembly and depends only on the distribution of masses in the assembly

and not upon the direction or magnitude ofω.

A more realistic situation obtains if a continuous rigid body is considered. In

this case,m(α)must be replaced everywhere byρ(r)dx dy dzand all summations

by integrations over the volume of the body. Written out in full in Cartesians,

the inertia tensor for a continuous body would have the form

I=[Iij]=





∫

(y^2 +z^2 )ρdV −

∫

xy ρ dV −

∫

xzρ dV

−

∫

xy ρ dV

∫

(z^2 +x^2 )ρdV −

∫

yzρ dV

−

∫

xzρ dV −

∫

yzρ dV

∫

(x^2 +y^2 )ρdV



,

whereρ=ρ(x, y, z) is the mass distribution anddV stands fordx dy dz;the

integrals are to be taken over the whole body. The diagonal elements of this

tensor are called themoments of inertiaand the off-diagonal elements without the

minus signs are known as theproducts of inertia.

Show that the kinetic energy of the rotating system is given byT=^12 Ijlωjωl.

By an argument parallel to that already made forJ, the kinetic energy is given by

T=^12

∑

α

m(α)

(

̇r(α)· ̇r(α)

)

=^12

∑

α

m(α)ijkωjx(kα)ilmωlx(mα)

=^12

∑

α

m(α)(δjlδkm−δjmδkl)x(kα)x(mα)ωjωl

=^12

∑

α

m(α)

[

δjl

(

r(α)

) 2

−x(jα)x(lα)

]

ωjωl

=^12 Ijlωjωl.

Alternatively, sinceJj=Ijlωlwe may write the kinetic energy of the rotating system as
T=^12 Jjωj.

The above example shows that the kinetic energy of the rotating body can be

expressed as a scalar obtained by twice contractingωwith the inertia tensor. It

also shows that the moment of inertia of the body about a line given by the unit

vectornˆisIjlˆnjnˆl(ornˆTInˆin matrix form).

SinceI(≡Ijl) is a real symmetric second-order tensor, it has associated with it

three mutually perpendicular directions that are itsprincipal axesand have the

following properties (proved in chapter 8):

(i) with each axis is associated a principal moment of inertiaλμ,μ=1, 2 ,3;

(ii) when the rotation of the body is about one of these axes, the angular

velocity and the angular momentum are parallel and given by

J=Iω=λμω,

i.e.ωis an eigenvector ofIwith eigenvalueλμ;