Let $(M,\omega)$ be a symplectic manifold and $\gamma:S^1 \rightarrow M$ be a contractible smooth loop. Is it possible to find an open set $U \subset M$ such that $\gamma(S^1) \subset U$ and such that there exists a Darboux chart $\phi : U \rightarrow \mathbb{R}^{2n}$?

Clearly this isn't true if $\gamma$ is not assumed contractible (there might not be any chart on $M$ that contains $\gamma(S^1)$!). If $\gamma$ is contractible then there certainly do exists charts that contain $\gamma(S^1)$, but do there necessarily exist *Darboux* charts?