Aircraft planar trajectories in crosswind navigation: some hypergeometric solutions

We study the planar motion of a self-propelled object against a withstanding flow, such as wind or sea current, whose speed is capable of perturbing the trajectory of the object, heading it off its path. Seven cases are dealt with, corresponding to fairly general wind types and all leading to nonlinear ordinary differential equations for the object motion. The first two cases concern motions described in Cartesian coordinates. They are followed by motions under a cyclonic wind, analysed in a polar reference frame; the wind is modelled as a logarithmic or hyperbolic spiral and closed-form solutions are obtained by means of the Gauss hypergeometric function and the Lerch transcendent. Further wind instances, treated in a Cartesian or in a polar reference system, require some special functions, such as the Lambert function, while finding the motion time law leads to a trinomial equation, in the most interesting situation of a mobile object that is faster than the wind. In the last case of a horizontal hyperbolic wind, time law and trajectory are computed in a polar reference frame by means of the Lambert function.