Timeline for References for the Poincaré-Cartan forms
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12 events
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| Sep 6, 2015 at 1:49 | comment | added | John Baez | I believe there's a typo in this answer, with "almost tangent structure" a typo for "almost complex structure". I can edit it... | |
| May 16, 2012 at 18:42 | comment | added | Robert Bryant | @Bonne: Well, presumably it wasn't called that until the last part of the 20th century. However, it certainly did exist much earlier in some form. Since my previous comment, I managed to peek into Giaquinta-Hildebrant, Volume 1, where, on page 396, they have a note that attributes the form $\omega$ that I gave in the previous comment to Beltrami (apparently in some work that he did in the 1860s). They also say that Hilbert rediscovered the form in 1900. They don't mention Poincaré, but I'd be surprised if it isn't there as well. | |
| May 16, 2012 at 15:31 | comment | added | Richard Bonne | All right. The strange thing is that my colleague had told me that the Poincaré-Cartan form was invented after the mid-20th century, so I can definitely say that he is wrong. | |
| May 15, 2012 at 23:13 | comment | added | Robert Bryant | @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. | |
| May 15, 2012 at 19:18 | comment | added | agt | @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. | |
| May 15, 2012 at 19:10 | history | edited | agt | CC BY-SA 3.0 |
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| May 15, 2012 at 17:50 | comment | added | Richard Bonne | @Bryant So you can confirm that Poincaré-Cartan forms was known (in it's modern form) before the twentieth century? | |
| May 15, 2012 at 17:14 | comment | added | Robert Bryant | 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. | |
| May 15, 2012 at 16:50 | comment | added | Richard Bonne | @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 (?). | |
| May 15, 2012 at 16:48 | history | edited | agt | CC BY-SA 3.0 |
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| May 15, 2012 at 16:35 | history | edited | agt | CC BY-SA 3.0 |
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| May 15, 2012 at 16:21 | history | answered | agt | CC BY-SA 3.0 |