Let $M^3$ be a Riemannian manifold with nonempty boundary and let $\Sigma$ be a smooth surface with boundary. Consider $\Phi : \Sigma \times (-\varepsilon, \varepsilon) \to M$ a proper variation of a fixed immersion $\varphi : \Sigma \to M$. This means that each map $\varphi_t(\cdot) = \Phi(\cdot, t)$ is an immersion with $\varphi_t(\partial \Sigma) = \varphi_t(\Sigma) \cap \partial M$ and $\varphi_0 = \varphi$.
If we denote the area element of $\Sigma$ in the metric induced by $\varphi_t$ by $dA_t$, the area functional of the variation is
$$A(t) = \int_{\Sigma} \, dA_t.$$
Then, it is well known that
$$A'(0) = - \int_{\Sigma} 2H f \, dA + \int_{\partial \Sigma} \langle \xi, \nu \rangle \, dL,$$
where $H$ is the mean curvature of the immersion $\varphi$, $\xi = \frac{\partial \Phi}{\partial t}\vert_{t=0}$ is the variational vector field, $f = \langle \xi, N \rangle$, with $N$ the unit normal for $\Sigma$ along $\varphi$, and $\nu$ the outward pointing unit normal for $\partial \Sigma$ in $\Sigma$. This is called the first variation of area.
I would like to know if there is an analogous formula for the first variation of the length of the boundary functional $L(t)$, given by
$$L(t) = \int_{\partial \Sigma} \, dL_t, $$
where $dL_t$ is the length element of $\partial \Sigma$ in the metric induced by $\varphi_t$. That is, is there a formula for $L'(0)$ involving known geometric quantities?