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Let $(M,\omega)$ be a symplectic manifold. Let $f:M \to \mathbb R$ be a smooth function. We have vector fields $X_f$ defined by $\omega(X_f,)=df$. Let $\phi_t$ be the flow of $X_f$ and let $\gamma: \left[a,b\right] \to M$ be a smooth curve. We define $\psi(t,s)=\phi_t(\gamma(s))$. How do we compute $$\int_a^b \int_0^1 \psi^*\omega$$

I use the following expression for $ \psi^*\omega$, $$ \psi^*\omega=\omega(\partial_s \psi ds + \partial_t \psi dt,\partial_s \psi ds + \partial_t \psi dt) $$ I suppose $\partial_t \psi = X_f(\phi_t(\gamma(s)))$ and $\partial_s \psi = D\phi_t \circ (d\gamma(s)/ds) $ but I don't know what to do after. I am not allowed to assume that $\phi_t$ preserves $\omega$.

vonbrand
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fidy
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  • I don't understand what you are trying to do, specifically. Are you trying to compute this explicitly? for that to make sense, it seems to me that you need some more explicit information. Or are you trying to do something else? this isn't clear to me. – Sam Lisi Apr 16 '13 at 09:26

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