A circular hoop rolling without slipping on a half-cylinder - by Lagrangian mechanics

In this video, we analyze the motion of a circular hoop released from the top of a half-cylinder fixed to the ground and rolling without slipping on its surface by using the Lagrangian mechanics. We try to find the angle where the circular hoop leaves the cylindrical surface by solving the constrained Euler-Lagrange equations.

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Background music:
0:00-6:39 Waltz of the Flowers (by Tchaikovsky) by Tchaikovsky
Creative Commons — Attribution 3.0 Unported— CC BY 3.0

6:39 - 8:55 No.1 A Minor Waltz - Esther Abrami (from YouTube Audio Library)

[uniscience-marwanmasrat]

Why is there a negative sign in equation 4?

[MahmoudHallak-cm]

you're goated wity the sauce my doggy

[akhilanr]

This is too good, helped me a lot, thank you, Your channel has needs a lot more subscribers!

[lazeau]

This was very helpful, I've been stuck on this problem for days! However, I would have liked an explanation for why the second constraint is the way it is. Several classmates and I have been debating why alpha(theta + phi) = R*theta instead of just alpha*phi = R*theta. I'm thinking it's because the "rolling without slipping" condition tells us that, as the hoop rolls along the cylinder, it traces out arc length alpha*phi which is equal to R*theta. Additionally, then, the hoop moving down along the cylinder traces out arc length alpha*theta, and thus we get the constraint you had written. This is still counterintuitive to me, though, and just generally feels odd.

[stephenrklein]

I believe your constraint equation (ii) is off. (R+a)theta_dot is the speed of the center of mass of the hoop, whereas a*phi_dot is the speed of the edge of the hoop, which aren't necessarily equivalent. Instead, shouldn't you have R*theta = a*phi, which would be the arc lengths traced out by the motion? The time derivatives of the two angles would follow directly from that constraint as well

[sayanjitb]

How did you know that \lambda_1 is the normal force?

[rabeyabosri]

I am not quite sure why you exchanged lambda1 and lambda2 at equations 5, 6 and 7. Lambda 2 has no dot derivatives

[rabeyabosri]

I don't know how you get that equation 4 🤣🤣🤣

[davidandresrodriguez]

Why is there a negative sign in equation 4?