Timeline for Symplectic submanifolds and first integrals
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| May 10, 2012 at 15:40 | comment | added | Vladimir S Matveev | Guiseppe, I do not know what to tell you. You have asked whether you are missing something -- no, you are missing nothing and your explanation is more precise than mine one.One can make mine explanation also mathematically precise (because your can build darboux coordinate system such that the differential of the first two coordinates coincide with $dC_1$ and $dC_2$ provided the poisson bracket of $C_1$ and $C_2$ is zero at a point.) This observation will be enough to justify my answer, and you do more or less the same in your answer. | |
| May 8, 2012 at 17:13 | comment | added | agt | But in your third paragraph, we need to consider even eventual isolated zeroes for $\{C_1,C_2\}=\Lambda(dC_1,dC_2).$ E.g. points $p\in\mathcal{N}$ such that $\{C_1,C_2\}(p)=0$ but $\{C_1,C_2\}(q)\neq 0,\ \forall q\in\mathcal{N}\setminus\{p\},$ And in such a case the same holds for $\{\overline{C}_1,\overline{C}_2\}$ independent from the choice of $\overline{C}_1,\overline{C}_2$). I hope to have clearly expressed my doubt, what I am missing? | |
| May 8, 2012 at 16:49 | comment | added | Vladimir S Matveev | Dear Guiseppe, you are right. To construct the Darboux coordinates we indeed need that the bracket of $C_1$ and $C_2$ vanish in a small neighborhood. Would the bracket of $C_1$ and $C_2$ vanish at the points of $\mathcal{N}$ only, one can find Poisson-commute functions $\tilde C_1$ and $\tilde C_2$ whose differentials at the point we consider coincides with that of $C_1$ and $C_2$, and construct Darboux coordinates starting from $\tilde C_1$ and $\tilde C_2$, which would be sufficient. | |
| May 8, 2012 at 15:44 | comment | added | agt | Dear Vladimir S Matveev I have a silly doubt about your third paragraph. Let be $p\in\mathcal{N},$ in order to find coordinates $x_1,\ldot,x_n,y_1,\ldots,y_n$ around $p$ in $\mathcal{M}$ with $\omega=dx_i\wedge dy_i$ and such that $x_1=C_1,x_2=C_2$ we need that $\{C_1,C_2\}$ vanishes not only at $p,$ but in a neighborhood of $p$ in $\mathcal{M}.$ At least this is my comprehension of the the Caratheodory-Jacobi-Lie theorem. I would highly appreciate if could explain to me what I am missing. Thank you. | |
| May 4, 2012 at 11:45 | history | edited | Vladimir S Matveev | CC BY-SA 3.0 |
added 7 characters in body
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| May 3, 2012 at 20:10 | history | answered | Vladimir S Matveev | CC BY-SA 3.0 |