Let $(M,\omega = d\alpha)$ be an exact symplectic manifold and $\gamma \in C^\infty(I,M)$ a path in $M$. Given $X_1,X_2 \in \mathfrak{X}(\gamma)$ I would like to compute
$$\frac{\partial^2}{\partial\sigma_1\partial\sigma_2}\bigg\vert_{\sigma_1 = \sigma_2 = 0}\gamma_{\sigma_1,\sigma_2}^* \alpha,$$ where $\gamma_{\sigma_1,\sigma_2}$ is a smooth $2$-parameter family of curves such that $\partial_{\sigma_1}\vert_{\sigma_1 = 0} \gamma_{\sigma_1,\sigma_2} = X_1$ and $\partial_{\sigma_2}\vert_{\sigma_2 = 0} \gamma_{\sigma_1,\sigma_2} = X_2$. By the superb answer of @TobiasDiez [here][1] we can use the generalised Cartan formula to compute
$$\frac{\partial}{\partial \sigma_1}\bigg\vert_{\sigma_1 = 0}\left( \gamma^*_{\sigma_1,0}i_{\partial_{\sigma_2}\vert_{\sigma_2 = 0}\gamma_{\sigma_1,\sigma_2}}(\omega \circ \gamma_{\sigma_1,0}) + d\gamma_{\sigma_1,0}^*i_{\partial_{\sigma_2}\vert_{\sigma_2 = 0}\gamma_{\sigma_1,\sigma_2}}(\alpha \circ \gamma_{\sigma_1,0})\right).$$
Equivalently,
$$\frac{\partial}{\partial \sigma_1}\bigg\vert_{\sigma_1 = 0}\left(\omega(\partial_{\sigma_2}\vert_{\sigma_2 = 0}\gamma_{\sigma_1,\sigma_2},\dot{\gamma}_{\sigma_1,0}) + d(\alpha(\partial_{\sigma_2}\vert_{\sigma_2 = 0}\gamma_{\sigma_1,\sigma_2}))\right).$$
I am not sure how to proceed further. 


  [1]: https://mathoverflow.net/questions/341930/differential-of-the-rabinowitz-action-functional