See attached picture for problem.
The Attempt at a Solution
So I have found the lagrangian to be:
L = ½ma2(θ'2+[itex]\omega[/itex]2sinθ) - mgacosθ
I think this is correct but I have some questions on the further solution of the problem. First of all - what is meant by the constants of motion? Is that the total energy? I can see in a sense that it should be conserved since the lagrangian doesn't depend on time. On the other hand - why would the energy be conserved from an intuitive point of view- i.e. if the hoop is moving at constant angular velocity regardless of the azimuthal angle then that must mean someone is compensating for the extra kinetic energy needed to maintain constant angular speed at larger angles (0;1/4[itex]\pi[/itex]) - I don't see how you could phyiscally have the gain in potential energy to go into this work.
Overall I am a bit unsure of how to picture the situation. At first I thought of a ball lying on top of a ring that is rotating about the z-axis (because that's what it is rotating about right?)
but then that situation doesn't really constrain the mass to move on the hoop. Then I thought of the hoop as a torus in which the ball is free to move. Is this a better picture?
And furthermore - since there exists a solution where the ball remains stationary at another point than the bottom that must mean some force is opposing the force of gravity on the mass. What is that force. Is that the force upwards from the exterior of the torus?
And lastly, I don't understand why you don't have to take into account the initial conditions when finding the angular velocity for which that mass can remain stationary. Surely there must be a difference between the situation where the mass starts at the top and the bottom for instance?