(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
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Broadly which area in differential geometry deals with this (is it symplectic geometry?)
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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)
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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
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