Let $(X,\omega)$ be a connected symplectic manifold, possibly with boundary. Let $g_1, g_2: B(1) \to X$ be two balls in $X$. Is it true that if $\delta$ is sufficiently small, then there is an ambient isotopy from $g_1|_{B(\delta)}$ to $g_2|_{B(\delta)}$? To be clear, by this I mean a smooth family of symplectic maps $\Psi_t : X \to X$ such that $\Psi_0 = id$ and $\Psi_1(g_1) = g_0$.

If this is true, does anyone have a proof or a reference?

Thanks.