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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.

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2 Answers 2

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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.

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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.)

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  • $\begingroup$ @Bryant and Tortorella Thank you for your suggestions. However I forgot to specify that I would be interested to know who invented the forms of Poincaré-Cartan (?). $\endgroup$
    – Richard Bonne
    Commented May 15, 2012 at 16:50
  • $\begingroup$ There is an historical discussion in Giaquinta and Hildebrandt that is quite thorough. The answer is more complicated than can be described fully here, but, roughly, one can say that the modern versions of the Poincaré-Cartan form emerged in the work of various different people in the 19th century, and it was only gradually recognized that these different things were all essentially the same. Poincaré, Cartan and Hilbert all had early versions that we would recognize as precursors of the general versions that exist today. $\endgroup$
    – Robert Bryant
    Commented May 15, 2012 at 17:14
  • $\begingroup$ @Bryant So you can confirm that Poincaré-Cartan forms was known (in it's modern form) before the twentieth century? $\endgroup$
    – Richard Bonne
    Commented May 15, 2012 at 17:50
  • $\begingroup$ @Richard Bonne I think that the idea of integral invariants was object of study at the end of 19th century before that the (now familiar) concepts of differential forms, differential manifolds were conceived. In the works of Poincarè should be possible met on one side the analysis of the motivating mechanical problems and on the other side the first depiction of the geometric notion of manifold, differential forms, and so on. $\endgroup$
    – agt
    Commented May 15, 2012 at 19:18
  • $\begingroup$ @Bonne: Well modern form is kind of in the eye of the beholder. Throughout the latter part of the 19th century, one finds in the mechanics literature, associated to a particle Lagrangian $L(x,y^i,p^i) dx$ (where $p^i$ stands for $dy^i/dx$ as usual), the Pfaffian expression $\omega = L\ dx + L_{p^i}\ (dy^i - p^i dx)$ (sum on $i$) in some form or another, and this is what we now call the Poincaré-Cartan form in the case of a first-order particle Lagrangian. I'm sure that the case of more independent variables can also be found, but I don't know any specific references off the top of my head. $\endgroup$
    – Robert Bryant
    Commented May 15, 2012 at 23:13

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