Let $\gamma \colon [0,\alpha] \to \mathbb{R}^{3}$ be a smooth regular curve and let $N\gamma$ denote its normal bundle.
Recall that $N\gamma$ is a smooth vector bundle, whose fiber at $\gamma(t)$ is the orthogonal complement $\dot{\gamma}(t)^{\perp}$ in $T_{\gamma(t)}\mathbb{R}^{3} \cong \mathbb{R}^{3}$ of the tangent vector $\dot{\gamma}(t)$.
For any smooth section $V$ of $N\gamma$, we define $D_{t}V$ to be the orthogonal projection onto $N\gamma$ of the Euclidean acceleration $\overline{D}_{t}V \equiv \dot{V}$.
Question: Given a smooth function $f \colon I \to \mathbb{R}$, does the linear ODE system on $N\gamma$ $$ \begin{cases} D_{t}V = f W\\ D_{t}W = -fV \end{cases} $$ have unique global solution for any initial condition $(v,w)$?