872 APPENDIX B. A RIGID BODY ROTATING ABOUT A POINT

In terms of the material basis, {e1 (t) ,e2 (t) ,e3 (t)} which is fixed with the body,

(ω (t)×x(t,x)) =

∣∣∣∣∣∣∣e1 (t) e2 (t) e3 (t)ω1 (t) ω2 (t) ω3 (t)

x1 x2 x3

∣∣∣∣∣∣∣where the ω i are the components of ω taken with respect to {e1 (t) ,e2 (t) ,e3 (t)} and aswe observed earlier, {x1,x2,x3} are the coordinates of the vector x(t,x) taken with respectto the {e1 (t) ,e2 (t) ,e3 (t)} . To simplify the integrand in 2.1 that long cross product issimplified.

Lemma B.0.3 Let a,b,c be three dimensional vectors. Then

a×(b×c) = (a ·c)b− (a ·b)c.

Proof: Let an orthonormal right handed coordinate system {e1,e2,e3} be given. Then

a×(b×c) = ε i jka j (b×c)k ei

= ε i jkεkpqa jbpcqei

= εki jεkpqa jbpcqei

= (δ ipδ jq−δ jpδ iq)a jbpcqei

= (a jbic j−a jb jci)ei

= (a ·c)b− (a ·b)c. ■

Now simplify the integrand using this lemma.

x(t,x)× (ω (t)×x(t,x))

= (x(t,x) ·x(t,x))ω (t)− (x(t,x) ·ω (t))x(t,x) .

Writing x(t,x) and ω (t) in terms of the material coordinates,

ω (t) = ω1e1 (t)+ω2e2 (t)+ω3e3 (t) ,

x(t,x) = x1e1 (t)+ x2e2 (t)+ x3e3 (t) ,

and sox(t,x)× (ω (t)×x(t,x)) =

∑i|x|2 ω iei (t)−

(∑

i∑

jx jω jxiei (t)

). (2.2)

Thus, listing the components of x(t,x)× (ω (t)×x(t,x)) with respect to the materialbasis yields the following in which x(t,x)×(ω (t)×x(t,x)) is written as a column vector.

(x2

1 + x22 + x2

3)

ω1−(x2

1ω1 + x2x1ω2 + x3x1ω3)(

x21 + x2

2 + x23)

ω2−(x2x1ω1 + x2

2ω2 + x3x2ω3)(

x21 + x2

2 + x23)

ω3−(x3x1ω1 + x3x2ω2 + x2

3ω3)