Hello. In my textbook by Jose Saletan called Classical Dynamics: A Contemporary Approach the author talks about TQ, the domain of the Lagrangian. He states that the space tangent to a point on the configuration manifold is in the tangent bundle, and that the entire tangent bundle can be thought of as just this applied to all points on the configuration manifold (if I'm understanding what he's saying correctly). He also states that the tangent bundle is where the veloicities of the system lie. He then goes on to give a concrete example: where the configuration manifold is a circle. He states that the tangent bundle then will be a cylinder. I don't really understand why this is. Clearly all the velocity vectors will lie in the plane of the circle. Why should the tangent bundle have components perpendicular to the circle?