References for the Poincaré-Cartan forms Hello, everybody. I'm looking for some reference about the Poincaré-Cartan form, I do not know how it is defined, I just know that it is used in Lagrangian mechanics but I have not found any references. Can you advise me a book that deals with the topic?
In particular I would be interested to know who invented the forms of Poincaré-Cartan.
Thank you very much.
 A: You could try the introductory sections of our book, "Exterior Differential Systems and Euler-Lagrange Partial Differential Equations" (authors Bryant, Griffiths, Grossman), which is available at http://arxiv.org/abs/math/0207039.  There are definitions, discussions, and references to the older literature there.
There is also considerable material in Giaquinta and Hildebrandt "Calculus of Variations", but it's not near the front.
Another place you can look is in the works of Ian Anderson and/or Peter Olver and their students.
A: I think that you could appreciate "Methods of Differential Geometry in Analytical Mechanics" by P.Rodriguez and M.deLeon (this is a link) 
Apart from its intrinsec interest as reference for both the constructions of differential geometry and the geometrization of Lagrangian/Hamiltonian mechanics, in particular, if you are looking for the role played by the Poincaré-Cartan forms in mechanics, you could find it in their Chapters 2 (for the canonical almost-tangent structure on $TQ$) and 9 (for the Lagrangian Mechanics).
There you would find that to any (Lagrangian) function $L$ on $TQ,$ there are associated the Poincaré-Cartan forms $\theta_L:=J^\ast dL$ and $\omega_L:=d\theta_L,$ where $J:T(TQ)\to T(TQ)$ is the vertical endomorphism associated to the canonical almost tangent structure on $TQ.$  
When the Lagrangian is not degenerate then $\omega_L$ is non degenerate, and the Euler-Lagrange vector field $\xi_L$ is the hamiltonian vector field w.r.t. $\omega_L$ with Hamilton function $E_L:=\mathcal{L}_{\Delta}L-L.$  (Here $\Delta\in\mathfrak{X}(TQ)$ is the Liouville vector field, i.e. the infinitesimal generator of the action of $\mathbb{R}$ throgh fiber-wise homotopies.) 
About the edited question: Truly my historical knowledge is very limited, but probably the origin of these concepts could be in the works by Poincarè on the celestial mechanics, (when the concept itself of differential forms was germinating,) and therefore at the origins of the differential topology as we know it nowadays. You could look at the EoM entry on integral invariants and the references therein. (This is a link.)
