# Regarding Legendre transform from tangent bundle to cotangent bundle

(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

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

2. 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)

3. 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)

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1. Kind of yes, symplectic geometry 'has it's origins in the Hamiltonion formuation of classical mechanics' according to wikipedia. It is, however, a branch of mathematics which has grown beyond that, to my understanding, and might be a too general search term for what you are looking for.
2. This I would not want to answer.
3. It is my impression that you are not looking for a textbook for phyicists but for a book aimed at mathematicians with an interest in theoretical physics. In that case W.D. Curtis and F.R. Miller, 'Differentiable Manifolds and Theoretical Phyisics' may be what you are looking for. It has a more differential geometric point of view than, e.g., V.I. Arnolds 'Mathematical Methods of Classical Mechanics', which stresses the ODE point of view. Volume II of Dubrovin, Fomenko and Novikovs 'Modern Geometry -- Methods and Applications' also covers the topic, but you have to work through quite some material before you reach that section. There are many other good books, and probably some newer ones, too, but I'd suggest you have a look at Curtis & Millers book if it available to you to see whether it meets your needs.
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Thank you! I will have a look at Curtis & Millers. But you would not want to answer 2. because you are not sure or for some other reason? (Thanks again) – Hsirihs Jun 18 '12 at 14:04
@Hsirihs This was never really my area of expertise. I learned this long ago (exam level), never really worked with it and forgot most of the details. So I know some books dedicated to the topic and that's about it. – user20266 Jun 18 '12 at 14:07

I would suggest Frankel

http://www.amazon.com/The-Geometry-Physics-An-Introduction/dp/1107602602/ref=pd_bxgy_b_img_c

which introduces the subject in light of physical applications (and less rigor, and very readable)

The answer to 2) is called "Fiber derivative" and usually not treated in introductory books.

http://en.wikipedia.org/wiki/Fiber_derivative

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