Drag is a mechanical force that characterizes the resistance that the fluid (liquid or gas) offers to the moving vehicle and acts in the opposite direction to its motion. The magnitude of the drag is typically proportional to the square of the speed and the power needed to overcome the drag is proportional to the cube of the speed.

In this talk we are interested in determine optimal trajectories of a vehicle moving in a fluid environment that minimize not only the power needed to overcome changes in velocity but also drag forces. Therefore, a variational problem where the energy functional depends on the acceleration and drag is formulated on the general context of a Riemannian manifold and the corresponding Euler-Lagrange equations are derived.

We will see that the presence of the drag force will increase substantially the complexity of the problem even when the geometry of the configuration space is not take into consideration. A numerical optimization approach will be presented in order to obtain approximate solutions for this problem in some particular configuration spaces.

In the absence of the drag, the problem boils down to the classical problem of finding a geometric cubic polynomial prescribing initial and final positions and velocities.