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I'm running into a functional associated to a piecewise smooth curve $\gamma: [0,1] \to V$, where $V$ is a real vector space with a symplectic form $\omega$:

$$ \int_{0 \leq x \leq y \leq 1} \omega(\gamma'(x), \gamma'(y))\ dx\ dy $$

Is this a standard concept? In particular, does it have a name?

In my application, $\gamma$ is something like the path from SW to NE under a partition, and this functional is the area of the partition. (Though generally I'm working in higher dimension.)

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    $\begingroup$ 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. $\endgroup$
    – Robert Bryant
    Commented Jun 17, 2017 at 2:58
  • $\begingroup$ 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! $\endgroup$
    – Allen Knutson
    Commented Jun 17, 2017 at 11:14
  • $\begingroup$ 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. $\endgroup$
    – Robert Bryant
    Commented Jun 17, 2017 at 11:38

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