The problem goes by this: A sphere of radius ##\rho## is constrained to roll without slipping on the lower half of the inner surface of a hollow cylinder of inside radius R. Determine the Lagrangian function, the equation of constraint, and Lagrange's equations of motion. Find the frequency of small oscillations. What I am thinking: Problem: I do not know how to relate the distance traveled by the center of mass of the sphere inside the cylinder. The sphere has a constraint related to its center of mass and its diameter. If it was in a plane it would be that the distance traveled its equal to the radius times an angle: ## x = \rho\times\theta##. However, the sphere is inside the cylinder... The sphere is rolling, thus it has kinetic energy. My question is: should I used the time derivative of the angle that I used to relate the motion of the sphere ? Any help would be appreciated. Thank you!