1. Hour 30 Euler’s Equations
Physics 321
Hour 30
Euler’s Equations

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. Relating Coordinates Start with 𝐿 in body coordinates
𝑒 𝑒 𝑒 𝐿 1 𝐿 2 𝐿 3 = 𝑒 𝑒 𝑒 𝐼 𝐼 𝐼 𝜔 1 𝜔 2 𝜔 3
= 𝑒 𝑒 𝑒 𝐼 11 𝜔 1 𝐼 22 𝜔 2 𝐼 33 𝜔 3

4. Relating Coordinates Transform 𝑑 𝐿 /𝑑𝑡 to space coordinates
𝑑 𝐿 𝑑𝑡 𝑠𝑝𝑎𝑐𝑒 = 𝑑 𝐿 𝑑𝑡 𝑏𝑜𝑑𝑦 + 𝜔 × 𝐿 𝑏𝑜𝑑𝑦
𝑥 𝑦 𝑧 𝐿 𝑥 𝐿 𝑦 𝐿 𝑧 = 𝑒 𝑒 𝑒 𝐿 𝐿 𝐿 3
+ 𝑒 𝑒 𝑒 3 𝜔 1 𝜔 2 𝜔 3 𝐿 1 𝐿 2 𝐿 3

5. Relating Coordinates
𝑥 𝑦 𝑧 Γ 𝑥 Γ 𝑦 Γ 𝑧 = 𝑒 𝑒 𝑒 𝐿 𝐿 𝐿 𝑒 𝑒 𝑒 3 𝜔 1 𝜔 2 𝜔 3 𝐿 1 𝐿 2 𝐿 3
Real external torque
In space coordinates
“Perceived” body torque

6. Relating Coordinates Γ 1 = 𝐼 11 𝜔 1 − 𝐼 22 − 𝐼 33 𝜔 2 𝜔 3
𝑒 𝑒 𝑒 Γ 1 Γ 2 Γ 3 = 𝑒 𝑒 𝑒 𝐿 𝐿 𝐿 𝑒 𝑒 𝑒 3 𝜔 1 𝜔 2 𝜔 3 𝐿 1 𝐿 2 𝐿 3
Real external torque
In body coordinates
“Perceived” body torque
Γ 1 = 𝐼 11 𝜔 1 − 𝐼 22 − 𝐼 33 𝜔 2 𝜔 3

7. Euler’s Equations Γ 1 = 𝐼 11 𝜔 1 − 𝐼 22 − 𝐼 33 𝜔 2 𝜔 3
Γ 1 = 𝐼 11 𝜔 1 − 𝐼 22 − 𝐼 33 𝜔 2 𝜔 3
Γ 2 = 𝐼 22 𝜔 2 − 𝐼 33 − 𝐼 11 𝜔 3 𝜔 1
Γ 3 = 𝐼 33 𝜔 3 − 𝐼 11 − 𝐼 22 𝜔 1 𝜔 2
It’s usually very hard to find the actual torques in terms of the rotating body coordinates!

8. 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
We can do these!

9. Example
Rotating a book - eulerseqs.nb

10. The Method of Ellipsoids
𝐼 11 = 𝑚 12 𝑏 2 + 𝑐 2 , 𝐼 22 = 𝑚 12 𝑎 2 + 𝑐 2 , 𝐼 33 = 𝑚 12 𝑎 2 + 𝑏 2
Letting 𝑎>𝑏>𝑐, then 𝐼 11 < 𝐼 22 < 𝐼 33
𝐿 2 = 𝐼 𝜔 𝐼 𝜔 𝐼 𝜔 3 2
𝑇= 1 2 𝐼 11 𝜔 𝐼 22 𝜔 𝐼 33 𝜔 3 2

11. The Method of Ellipsoids
These are two ellipsoids with equations
𝜔 𝐿 𝐼 𝜔 𝐿 𝐼 𝜔 𝐿 𝐼 =1
𝜔 𝑇 𝐼 𝜔 𝑇 𝐼 𝜔 𝑇 𝐼 =1
Note that the ellipsoids are in ω-space. They don’t predict motion. Possible values of ω are given by the intersection of the ellipsoids.

12. Example
ellipsoids.nb

13. 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

14. Euler’s Equations – No Torques, I11=I22 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

15. In the space axes 𝐿 = 𝐿 𝑧 𝑧 and:
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

16. Example
football.nb