Geometric point of view of configuration space and Lagrangian mechanics 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:
K. Sundermeyer's Constrained Dynamics states on p. 32:

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 $TTQ.$

 A: Firstly, it is important to point out that the configuration space need not be a tangent bundle. The configuration space is the manifold of "positions". What is a tangent bundle is the velocity phase space, comprised of the points of the form $(q, \dot{q})$ (I'll omit indices for simplicity of notation).
I'll try to describe the general idea by roughly following Sundermeyer's argument. We start with a configuration space, the possible "positions" (or, well, configurations) that the system might assume. For example, for a simple pendulum, the configuration space is a circle, since the pendulum is constrained to stay at a fixed distance from the origin in a given plane. Apart from these considerations, the pendulum can move freely, so the total collection of positions it might assume forms a circle.
Now, we want to describe the dynamics of this thing. However, using just the positions one can't do that. For example, if you want to compute the energy of the system, you need to know its velocity as well. If the pendulum is at its lowest point, it can still have very different trajectories depending on how fast it is moving. Therefore, if we want to actually describe the state of a system, we'll need to also know the velocities.
In a manifold $Q$, the possible velocities at a point $p$ are simply all of the vectors defined at $p$, i.e., all of $T_p Q$. This is pretty much the intuitive notion that, at $p$, one could be moving with an arbitrary speed at an arbitrary direction, as long as they are staying tangent to the manifold (otherwise, their movement would take them out of the configuration space, which is not possible, since the configuration space already comprises all possible positions). If we now being all of the $T_pQ$ spaces together, we get
$$TQ = \bigcup_{p \in Q} \lbrace p\rbrace \times T_pQ.$$
This is the tangent bundle. Notice the points of the tangent bundle are of the form $(p,v)$, where $p$ is some position and $v$ is some velocity. Each point in the tangent bundle now characterizes completely the state of movement of the system. If you choose a point $(p,v)$ for the pendulum, you have initial conditions which allow you to solve for its movement and completely compute how it will evolve.
In short, the configuration space is not a tangent bundle, the phase space is. That is essentially because one also needs to know the velocity to characterize a system's state of motion.
