Even though (I think) I understand the concept of a tangent bundle, I have trouble assimilating the idea of the configuration space being one and in relation to what that is the case. How can I understand that intuitively (but with formal basis, if possible)?
For instance, here's a generality I can't understand:
Sundermeyer (Constrained Dynamics, page 32) states:
The configuration space itself is unsuitable in describing dynamics [...]. One needs at least first order equations, and geometrically these are vector fields. So we have to find a space on which a vector field can be defined. An obvious candidate is the tangent bundle $TQ$ to $Q$, which may be identified with the velocity phase space. [...]. Lagrangian mechanics takes place on $TQ$ and $TTQ.$