Least crossing time of a boat in a time-dependent current.

The problem of minimizing the time required to cross a current-affected region from one shore to the other serves as a classical illustration of reference frame transformation in physics, as well as minimizing a functional with respect to trajectories and subject to boundary conditions. In the case where the current is uniform, constant and slower than the boat's speed, the optimal path corresponds to a straight-line trajectory, with the boat adopting a constant crab angle to counteract the drift induced by the current. When the current varies with position, the optimal trajectory requires to adjust the boat's nose angle accordingly with location. This optimization problem can be addressed using the calculus of variations, ultimately yielding the Euler–Lagrange equation as the governing condition. To the best of our knowledge, the case of a space-independent time-dependent current is seldom addressed in the literature. In the present study, we provide a straightforward demonstration that the trajectory minimizing the travel time corresponds to a constant boat angle, determined by the time-averaged value of the current, using only some simple properties of integrals. This general result is illustrated through a specific example in which the current increases linearly with time.