Page 207 - Intro to Tensor Calculus

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EXAMPLE 2.2-5. (Euler angles φ, θ, ψ) Consider the following sequence of transformations which
are used in celestial mechanics. First a rotation about the x 3 axis taking the x i axes to the y i axes
     
y 1 cos φ sin φ 0 x 1
 y 2  =  − sin φ cos φ 0   x 2 
y 3 0 0 1 x 3
where the rotation angle φ is called the longitude of the ascending node. Second, a rotation about the y 1
0
axis taking the y i axes to the y axes
i
     
y 0 1 1 0 0 y 1
 y 0  =  0 cos θ sin θ   y 2 
2
y 0 3 0 − sin θ cos θ y 3
where the rotation angle θ is called the angle of inclination of the orbital plane. Finally, a rotation about
0
the y axis taking the y axes to the ¯x i axes
0
i
3
     
¯ x 1 cos ψ sin ψ 0 y 0 1
 ¯ x 2  =  − sin ψ cos ψ 0   y 0 
2
¯ x 3 0 0 1 y 0 3
where the rotation angle ψ is called the argument of perigee. The Euler angle θ is the angle ¯x 3 0x 3 , the angle
φ is the angle x 1 0y 1 and ψ is the angle y 1 0¯x 1 . These angles are illustrated in the figure 2.2-6. Note also that
˙ ˙ ˙
the rotation vectors associated with these transformations are vectors of magnitude φ, θ, ψ in the directions
indicated in the figure 2.2-6.

Figure 2.2-6. Euler angles.

By combining the above transformations there results the transformation equations (2.2.50)
     
cos ψ cos φ − cos θ sin φ sin ψ cos ψ sin φ +cos θ cos φ sin ψ sin ψ sin θ
¯ x 1 x 1
 ¯ x 2  =  − sin ψ cos φ − cos θ sin φ cos ψ − sin ψ sin φ +cos θ cos φ cos ψ cos ψ sin θ   x 2  .
¯ x 3 sin θ sin φ − sin θ cos φ cos θ x 3
It is left as an exercise to verify that the transformation matrix is orthogonal and the components ` ji
satisfy the relations (2.2.51).