In this talk I will focus on the first variational formula for the length functional. First of all I will start by showing the geodesic equation for curves immersed in a Riemannian manifold. Then, I will describe the behavior of the length functional for curves immersed in a graded manifold equipped with a Riemannian metric. The first variation can be computed only if the curve can be deformed in a suitable sense. This condition is given by the surjection of the holonomy map that expresses the controllability of a differential system along the curve. This map gives us the possibility to define when a curve is regular or singular. Finally, I will show the Euler-Lagrange equation for immersed regular curves.