1. Physics 321
Hour 31
Euler’s Angles

2. Space and Body Coordinates
𝑒 3 𝑧
𝑒 2 𝑦 𝑥
𝑒 1
Body coordinates are on principal axes
If possible, use c.m. as origin in both frames
If not possible, c.m. motion is easy

3. Euler’s Equations – No Torques
𝐼 11 𝜔 1 = 𝐼 22 − 𝐼 33 𝜔 2 𝜔 3
𝐼 22 𝜔 2 = 𝐼 33 − 𝐼 11 𝜔 3 𝜔 1
𝐼 33 𝜔 3 = 𝐼 11 − 𝐼 22 𝜔 1 𝜔 2

4. Euler’s Equations – No Torques, I11=I22
𝐼 11 𝜔 1 = 𝐼 11 − 𝐼 33 𝜔 2 𝜔 3
𝐼 22 𝜔 2 = 𝐼 33 − 𝐼 11 𝜔 3 𝜔 1
𝐼 33 𝜔 3 =0
𝜔 3 is a constant
𝜔 1 = 𝐼 11 − 𝐼 33 𝐼 11 𝜔 2 𝜔 3 ≡ Ω 𝑏 𝜔 2
𝜔 2 =− 𝐼 11 − 𝐼 33 𝐼 11 𝜔 3 𝜔 1 ≡ −Ω 𝑏 𝜔 1
𝜔 2 =− Ω 𝑏 𝜔 1 =− Ω 𝑏 2 𝜔 2

5. Euler’s Equations – No Torques, I11=I22 In the body axes:
𝜔 = 𝜔 0 cos Ω 𝑏 𝑡 − 𝜔 0 sin Ω 𝑏 𝑡 𝜔 3
𝐿 = 𝐼 11 𝜔 0 cos Ω 𝑏 𝑡 − 𝐼 11 𝜔 0 sin Ω 𝑏 𝑡 𝐼 33 𝜔 3

6. In the space axes: Ω 𝑠 = 𝐿 𝐼 11
Ω 𝑠 = 𝐿 𝐼 11
𝜔 = 𝜔 0 sin α cos Ω 𝑠 𝑡 𝜔 0 sin 𝛼 sin Ω 𝑠 𝑡 𝜔 0 cos 𝛼
𝑒 3 = sin 𝜃 cos Ω 𝑠 𝑡 sin 𝜃 sin Ω 𝑠 𝑡 cos 𝜃
Prolate object: Ωb<0, Ωs>0

7. Example
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8. Constants of the Motion, No Torque
𝐿 = 𝐼 11 𝜔 0 cos Ω 𝑏 𝑡 − 𝐼 11 𝜔 0 sin Ω 𝑏 𝑡 𝐼 33 𝜔 3 in body, has constant length
Ω 𝑏 = 𝐼 11 − 𝐼 33 𝐼 11 𝜔 3
𝜔 3 is constant
𝐿 in space is constant
𝐿 𝑧 is constant
cos 𝜃= 𝐿 𝑧 /𝐿 so θ is constant
cos 𝛼= 𝜔 𝑧 /𝜔 so ω is constant

9. Example
HW31 Answers.nb

10. Unit Vectors 𝑒 ′ 3 = 𝑒 3 = sin 𝜃 cos 𝜑 𝑥 + sin 𝜃 sin 𝜑 𝑦 + cos 𝜃 𝑧
𝑒 ′ 1 = cos 𝜃 cos 𝜑 𝑥 + cos 𝜃 sin 𝜑 𝑦 − sin 𝜃 𝑧

11. Angular Velocities 𝜓 is spin about the body 3-axis
𝜃 is tipping of the body 3-axis
𝜑 is precession about the space z-axis
𝜔 = 𝜑 𝑧 + 𝜃 𝑒 ′ 2 + ψ 𝑒 3
𝑧 = 𝑒 3 cos 𝜃 − 𝑒 ′ 1 sin 𝜃
𝜔 =− 𝜑 sin 𝜃 𝑒 ′ 1 + 𝜃 𝑒 ′ 2 +( ψ + 𝜑 cos 𝜃 ) 𝑒 ′ 3

12. Angular Momentum 𝐿 =− 𝐼 11 𝜑 sin 𝜃 𝑒 ′ 1 + 𝐼 22 𝜃 𝑒 ′ 2
+ 𝐼 33 ( ψ + 𝜑 cos 𝜃 ) 𝑒 ′ 3
𝐿 3 = 𝐼 33 ψ + 𝜑 cos 𝜃
𝐿 𝑧 = 𝐿 ∙ 𝑧 = 𝐼 11 𝜑 sin 2 𝜃+ 𝐿 3 cos 𝜃
→ 𝜑 = 𝐿 𝑧 − 𝐿 3 cos 𝜃 𝐼 11 sin 2 𝜃
For torque-free systems, Lz and L3 are constants, so 𝜑 is also constant.

13. Kinetic Energy 𝑇= 1 2 𝐼 11 𝜔 1 2 + 1 2 𝐼 22 𝜔 2 2 + 1 2 𝐼 33 𝜔 3 2
𝑇= 1 2 𝐼 11 𝜔 𝐼 22 𝜔 𝐼 33 𝜔 3 2
= 1 2 𝐼 𝜑 2 sin 2 𝜃+ 𝜃 𝐼 ψ + 𝜑 cos 𝜃 2