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I posted this question here on math.stackexchange.com. I have not had an answer, and I thought it could be more appropriate here.
(Please, If you judge this my opinion is wrong, then I will delete this question)

Reading a paper I had the need to complete a proof, and came up with a certain argument(see below). My question is: at your knowledge, could I reduce it to a special case of some other theorem? I ask this question in order to give a correct reference, instead of my trivial ad hoc argument, in the case the answer is positive.

I had to prove that:
Given a smooth action $\Psi$ of $\mathbb{T}^k$ on a symplectic manifold $(M,\omega)$, if $\omega$ is exact and there exists a smooth map $\pi:M\to P$ constant on the orbit of $\Psi$ and such that $\zeta_X\lrcorner\omega\in\pi^*(\Omega^1(P)),\forall X\in\textrm{Lie}(\mathbb{T}^k)$,( being $\zeta$ the action of $\textrm{Lie}(\mathbb{T}^k)$ on $M$ induced by $\Psi$), then the $\Psi$ is an hamiltonian action w.r.t. $\omega$.


For completeness, I sketch also the trivial proof:
Let $\eta$ be a primitive of $\omega$, and $\mu_i\in C^\infty(M,\mathbb R)$ be defined by $$\mu_i=\int_0^1 \big(\textrm{Fl}_t^{\zeta_{e_i}}\big)^\ast (\zeta_{e_i}\lrcorner\eta)\,dt.$$ We contend that $d\mu_i=\zeta_{e_i}\lrcorner\omega$, for any $i=1,\ldots,k$.
By H.Cartan's formula and the theorem on Lie derivative we get $$d\mu_i=\int_0^1\frac{d}{dt}\bigg(\big(\textrm{Fl}_t^{\zeta_{e_i}}\big)^\ast\eta\bigg)\,dt-\int_0^1\big(\textrm{Fl}_t^{\zeta_{e_i}}\big)^\ast(\zeta_{e_i}\lrcorner\omega)\,dt.$$ The first integral is identically zero for periodicity.
The second one is $\zeta_{e_i}\lrcorner\omega$, because its integrand is a constant function of $t$, i.e. $\mathcal{L}(\zeta_{e_i}).(\zeta_{e_i}\lrcorner\omega)=0$ by the hypothesis.

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    $\begingroup$ @Theo Buehler: Yes surely it is due to the foundational work of Elié Cartan, but I thought it was named after his son Henri Cartan, for this formula is singled out in his axiomatic presentation of the equivariant cohomology for smooth manifolds acted by a Lie group. Precisely, I found it under this name in Symplectic Geometry and Analytical Mechanics of Libermann and Marle. $\endgroup$
    – agt
    May 29, 2011 at 20:37
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    $\begingroup$ @Giuseppe: The formula you are calling H. Cartan's formula first appeared, to my knowledge, in É. Cartan's 1922 book "Leçons sur les invariants intégraux", and these are lectures based on a course that he had given a couple of years earlier, so the formula might date from around 1920, when H. Cartan would have been about 16. $\endgroup$ May 29, 2011 at 22:25
  • $\begingroup$ @Robert Bryant: Thank you very much for the attention, and the careful historical information. I did not known exactly where, in the work of Elie Cartan, this formula appeared for the first time. Please, my previous observation was just an hypothesis to explain the attribution of this formula to Henri instead than to Elie, as I found it sometimes in the literature. $\endgroup$
    – agt
    May 30, 2011 at 6:13

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