Geometric point of view of configuration space and Lagrangian mechanics

Question

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 $TTQ.$

This post imported from StackExchange Physics at 2016-07-04 12:09 (UTC), posted by SE-user Mario Barela

Comments

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).

This post imported from StackExchange Physics at 2016-07-04 12:09 (UTC), posted by SE-user doetoe

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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.

This post imported from StackExchange Physics at 2016-07-04 12:09 (UTC), posted by SE-user Qmechanic