Timeline for A functional on paths in a symplectic vector space
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| S Jun 20, 2017 at 18:58 | history | suggested | Ali Taghavi |
I add two tags
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| Jun 20, 2017 at 18:47 | review | Suggested edits | |||
| S Jun 20, 2017 at 18:58 | |||||
| Jun 17, 2017 at 11:38 | comment | added | Robert Bryant | I'm sorry about being nit-picky; I thought that was what you meant, but I wasn't sure. I guess the phrase "Let $(V,\omega)$ be a symplectic vector space" would have removed all doubt for me about what you meant. (Also, the title of the question should have given me a clue.) I was also a little imprecise in my answer, since I should have written "integrating $\gamma^*\alpha$ over $[0,1]$, where $\alpha$ is the $1$-form on $V$ defined at each $v\in V$ by $\alpha_v(w) = \omega(v,w)$", etc. Btw, the only critical points of your functional are the constant maps. | |
| Jun 17, 2017 at 11:14 | comment | added | Allen Knutson | Yes, I was assuming constant coefficients; I'd hoped "vector space with a symplectic form" to mean that, as opposed to "symplectic manifold that happens to be diffeomorphic to a vector space". I'll see if I can make use of the 1-form. Thanks! | |
| Jun 17, 2017 at 2:58 | comment | added | Robert Bryant | Are you assuming that $\omega$ has constant coefficients (when expressed in linear coordinates on $V$)? If so, then this integral can be evaluated explicitly as integrating over $\gamma$ the 1-form $\alpha$ defined at each point as $\alpha_v(w) = \omega(v,w)$ and then then subtracting $\omega(\gamma(0),\gamma(1))$. Otherwise, your functional reminds me of K. T. Chen's iterated integrals. Maybe looking at those references will turn up the kind of thing you might be looking for. | |
| Jun 17, 2017 at 1:30 | history | asked | Allen Knutson | CC BY-SA 3.0 |