Timeline for The Jacobi Identity for the Poisson Bracket
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Apr 27, 2011 at 20:15 | comment | added | agt | @Josè Figueroa-O'Farrill: Thanks for the attention. In my answer, I tried to highlight this point that was already in your answer. But my approach is lowbrow with respect to the highbrow answer of Jonathan. | |
| Apr 26, 2011 at 3:53 | comment | added | José Figueroa-O'Farrill | @Giuseppe: you are correct. Even if $\omega$ is not closed, if you define the hamiltonian vector field $X_f$ for a smooth function $f$ by $\omega(X_f,Y) = Yf$ for all vector fields $Y$, then again you will find that $d\omega(X_f,X_g,X_f)$ is (perhaps up to a sign) the "Jacobiator" of $f,g,h$. | |
| Apr 24, 2011 at 16:57 | comment | added | agt | @Josè Figueroa-O'Farrill: When, for an arbitrary almost-symplectic manifold, we again construct the bracket, is correct that $d\omega(X_f,X_g,X_h)$ is equal to the Jacobiator $J(f,g,h)$? or I am making same mistake? | |
| Apr 24, 2011 at 7:23 | comment | added | Zack | An alternative after only proving $\iota_{[X_f,X_g]}\Omega=d\lbrace g,f\rbrace$ is to observe that the derivative of $\lbrace\lbrace f,g\rbrace h\rbrace+\lbrace\lbrace h,f\rbrace g\rbrace+\lbrace\lbrace g,h\rbrace f\rbrace$ is zero by the ordinary Jacobi identity. So it's locally constant. Since it's linear and locally defined, it must be zero (pick a point and kill it somewhere else with a bump function). | |
| Jun 21, 2010 at 23:00 | comment | added | José Figueroa-O'Farrill | There is no mistake in your calculations, as far as I can see. The last equation you write down is certainly correct. It simply requires extra work to derive the Jacobi identity from that point. I am afraid that the only way I can see to do this is basically to undo some of your identities and get back to something akin to what I wrote in my answer. | |
| Jun 21, 2010 at 20:47 | vote | accept | Paul Siegel | ||
| Jun 21, 2010 at 20:47 | comment | added | Paul Siegel | I had to take care of some other stuff before I could come back to this and work through the details, but everything seemed to check out. Thanks! I wonder where the mistake lay in my original approach, which used more or less the same tools. | |
| Jun 17, 2010 at 7:13 | history | edited | José Figueroa-O'Farrill | CC BY-SA 2.5 |
had missed a comma
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| Jun 16, 2010 at 20:07 | history | answered | José Figueroa-O'Farrill | CC BY-SA 2.5 |