How to calculate the first variation of length of a curve $\gamma?$ The length is defined as $$L(\gamma) = \int_{\gamma}ds.$$
So the first variation is $$\frac{d}{dc}L(\gamma + c\phi)|_{c=0} = \frac{d}{dc}\int_{\gamma + c\phi}ds\bigg|_{c=0} = \int_{\gamma +c\phi}\nabla_{\gamma + c\phi}\cdot v_c\bigg|_{c=0}$$ (where $v_c$ is the velocity of the curve $\gamma + c\phi$)
$$= \int_{\gamma +c\phi}v_c \cdot \kappa_c \nu_c\bigg|_{c=0}$$
where $\kappa_c$ and $\nu_c$ are the mean curvature and unit normal vector on $\gamma + c\phi.$
How can I relate the quantities [$v_c$, $\kappa_c$ and $\nu_c$] to [$v$, $\kappa$ and $\nu$] (the respective quantities on $\gamma$)? I don't think I can just evaluate the above on $c=0$ and remove the subscript $c$'s.
Edit: I'd prefer answers not to have any heavy Riemannian geometry stuff (so tensors and g^{ij} and stuff like that). Thanks. Also the reason I'm asking is to show that mean curvature is the gradient flow of the length integral so I want to see how they end up with $\int_\gamma{\kappa \phi}$ at the end of the above calculation.
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