(I'm a complete beginner at differential geometry)
I'm studying about constrained systems, in which we "map a Lagrangian system from a tangent to a cotangent bundle. Hamiltonian dynamics then appears as image dynamics via the Legendre map which is degenerate. A study of image of (hamiltonian) dynamics is possible if the Legendre map has constant rank."
More specifically from what I've (barely) understood, we have a configuration space $(q_1, ..., q_n, v_1, ..., v_n)$ or $(q, v)$ in short, where $v_i = dq_i/dt$, the config space being regarded as tangent bundle. Now we perform a legendre transform of lagrangian $L$. We obtain a map from the TB to the cotangent bundle $(q, p)$, where $p_i = ∂L/∂v_i$.
Now the rank of the Hessian matrix $∂L/(∂v_i∂v_j)$ is supposed to determine some property of the image of the map in the cotangent bundle, which I can't understand (intuitively I think it determines the image as a subset of the cot bundle, but that's a vague idea).
For me to better understand this, could someone please point out
Broadly which area in differential geometry deals with this (is it symplectic geometry?)
Which theorem(s)/result(s) precisely deals with whatever I've stated above (nature of Legendre transform and relation of rank of that Hessian matrix to the image in cotangent bundle)
Which book on differential geometry would be recommended that also treats the same area that I've asked about in 1., and also would be good as a first reading
I'm anyway going to study differential geometry, but only from the view of using it in higher-level Physics. So it would be highly helpful if it could further be mentioned which part/sections of the recommended book(s) I would have to read (ones that have applications in Physics)
Thanks in advance