Let $G: \mathbb{R}^2\longrightarrow\mathbb{R}^2$ be a smooth function. Let also $\Gamma:[0,1]\longrightarrow\mathbb{R}^2$ be a smooth curve.
I am working on the problem of finding a curve $\gamma$ such that $G\circ\gamma = \Gamma$.
I think that a possible method to find an approximation of $\gamma$ would be sampling points $(X_k,Y_k)$ on $\Gamma$, for instance $(X_k,Y_k)=\Gamma(f(k))$ for some suitable function $f$, work to find points $(x_k,y_k)$ such that $G(x_k,y_k)=(X_k,Y_k)$ and then find an interpolation curve that passes through thesee points. This could work when $G$ is injective but even then finding preimages is not an easy task.
Moreover, this strategy would be utterly cumbersome if $G$ is not injective.
My question is this: is there a known method of finding such a curve $\gamma$?
I am currently trying via variational calculus, setting \begin{equation} I[\gamma] := \int_0^1\|G(\gamma(t))-\Gamma(t)\|^2\ \text{dt} \end{equation} as the functional to minimize, but the resulting differential equations are incredibly difficult.
Note: even an effective hint for finding an approximation for $\gamma$ would be really helpful.