1. Example: A bead sliding on a uniformly rotating wire in force-free space
This is a trivial result:
In the rotating frame of reference, the bead experiences centrifugal acceleration
To find force of constraint we can use angular momentum:

2. Example: Bead on a spinning wire hoodθ R ρ ω
Method 1: using forces
In equilibrium:
Method 1: using Lagrangian
Two components of velocity:

3. Example: Bead on a spinning wire hood (Stability of equilibrium, and oscillations)1) 2)

4. Example: Pendulum whose point of support is forced to rotatey ϕ m l r

5. Example: Pendulum whose point of support is oscillating horizontallyϕ m l
x1=acos(ωt)

6. Proof of Lagrange’s equation with constraints