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:

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 $TTQ.$

• Your revision (v4) is an improvement in that you now have fewer distinct questions to discuss, but it's still quite broad. Perhaps you could some up a specific example where the tangent bundle/configuration space viewpoint seems like it would be useful, but you can't quite wrap your head around it; we could generalize from there. – rob Jun 29 '16 at 21:30
• I don't think the question in its present form is too broad at all. If you more or less know what a tangent bundle is, it can be seen or shown quite easily that the Lagrangian has to be a function on the tangent bundle of configuration space, and if you don't, you can't really define the Lagrangian (that doesn't mean you cannot derive correct physics from it). – doetoe Jun 30 '16 at 9:53
• I really don't see why the question would be considered too broad now. – GaloisFan Jun 30 '16 at 13:50
• Comment to the post (v6): It seems that Sundermeyer with the last sentence just wants to say that velocity and acceleration belong to $TQ$ and $TTQ$, respectively. – Qmechanic Jul 1 '16 at 8:25
• I suggest to look at the first 7 chapters in the book "Dynamical Systems a Differential Geometric Approach to Symmetry and Reduction" by Marmo, Saletan, Simoni, Vitale. Specifically, chapter 3 contains an extensive discussion of how to construct the configuration space from empirical data, and chapters 6 and 7 deal with the problem of obtaining differential equations from observed trajectories. In this case, the tangent bundle of the configuration space arises as the simplest case of what they call a lifting procedure for dynamical curves. – SepulzioNori Mar 13 at 13:44