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:

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

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

Example: Pendulum whose point of support is forced to rotate y ϕ m l r

Example: Pendulum whose point of support is oscillating horizontally ϕ m l x1=acos(ωt)

Proof of Lagrange’s equation with constraints