882 APPENDIX C. LAGRANGIAN MECHANICS
Here ω i are the components of ω taken with respect to the ei (t)= ei (t) . Thus, as in SectionB, |ω (t)×x(t,x)|2 =
ωT (t)
I11 I12 I13
I12 I22 I23
I13 I23 I33
ω (t)
Where
Ikk =∫
B(0)
(∑j ̸=k
x2j
)ρ (x1,x2,x3)dx
Ii j = −∫
B(0)xix jρ (x1,x2,x3)dx, i ̸= j.
As in this section, choose x1 (0) ,x2 (0) and x3 (0) such that Ii j = 0 whenever i ̸= j. There-fore, the kinetic energy in terms of the components of ω taken with respect to the axes,xi (t) is seen to be
T =12
3
∑k=1
Ikωk (t)2 .
Note that Ik is independent of t and the ωk are the components of ω taken with respect to theaxes, xi (t). While this is a nice formula, we want to relate it to the Euler angles because theEuler angles have more geometric significance. Therefore, what we need to find is ωk (t) interms of the time derivatives of the Euler angles. Refering to the above picture of the Eulerangles, we see that φ
′ contributes a term, to the angular velocity vector which is of theform (0,0,φ ′) where these are the components taken with respect to x1
1,x12 and x1
3. Writingthis vector in terms of the axes, x2
1,x22 and x2
3, we get (0,φ ′ sin(θ) ,cos(θ)φ′) . Now to this
we add the angular velocity vector contributed by θ′ which with respect to the axes, x2
1,x22
and x23 is
(θ′,0,0
). Therefore, in terms of x2
1,x22 and x2
3, we have the total angular velocityvector resulting from θ and φ is
(θ′,φ ′ sin(θ) ,cos(θ)φ
′) . Now we write this vector interms of the final coordinate system, x3
1,x32 and x3
3 = x1 (t) ,x2 (t) and x3 (t) . This yields(cos(ψ)θ
′+ sin(ψ)sin(θ)φ′,cos(ψ)sin(θ)φ
′− sin(ψ)θ′,cos(θ)φ
′) . To this we mustadd the contribution to the angular velocity from ψ ′ which in terms of this last system ofcoordinate axes is just (0,0,ψ ′) . Therefore, in terms of x1 (t) ,x2 (t) and x3 (t) we have theangular velocity is
ω =(cos(ψ)θ
′+ sin(ψ)sin(θ)φ′,cos(ψ)sin(θ)φ
′− sin(ψ)θ′,cos(θ)φ
′+ψ′) .
Therefore, the kinetic energy is
T =12
(I1(cos(ψ)θ
′+ sin(ψ)sin(θ)φ′)2
+
I2(cos(ψ)sin(θ)φ
′− sin(ψ)θ′)2
+ I3(cos(θ)φ
′+ψ′)2). (3.4)
Now we will consider a spinning top or gyroscope. Consider the following picture.There are two planes through the origin, one perpendicular to the x3 axis, and one perpen-dicular to the x3 (t) axis. They intersect in the line of nodes shown in the picture. Also, inthe above discussion of the Euler angles, we see the x1
1 axis is in the plane perpendicular tox3 and also is in the plane perpendicular to x3
3 = x3 (t) here. Therefore, φ is as shown in the