Relative equilibrium

• The unit time is (R/g)^1/2, where R is the ring radius.
• You can select ω and the initial angle θ (measured from the lowest ring point).
• The point of view of the three-dimensional projection can be changed with the mouse (or the sliders).
• Each whole image can be moved with the mouse while pressing Ctrl.
• To change the zoom in the projections, press Shift when moving up or down the mouse pointer.
• Put the mouse pointer over an element to get the corresponding tooltip.

Activities

1. Write down the Lagrangian and the equation of motion.
2. Discuss the existence and stability of configurations of relative equilibrium (in which the mass is at rest with respect to the ring) in function of the dimensionless parameter Ω = (R/g)^1/2 ω. (It may be helpful to consider an equivalent problem in the form m θ'' = -V'(θ), for an appropriate effective potential V.)
3. You should conclude that at Ω = 1 a bifurcation arises (i.e., the number and/or stability of equilibrium points change).
4. Use the simulation to check your conclusions.
5. Can you compute (and check) the frequency of small oscillations around stable equilibrium configurations?