Chasing relativistic rabbits (Q1893814)

The problem of determining the curve of a pursuer following a pursued object which is moving along a given path has a long history. The author looks at the problem from the viewpoint of special relativity; here the pursuer is always moving in the direction where he sees the pursued at the current instant of time, which is, of course, already in the classical case a rather inefficient strategy, especially since the pursued is assumed to follow a fixed path without regard to the threat the pursuer poses for him. Thus, the problem is to be seen as an exercise in nonlinear differential equations of second order rather than being of any relevance to the warfare-related pursuit problem. After formulating the problem and supplying the differential equations, the author looks at the particular cases where the velocities (with respect to a given inertial frame of reference) are constant and the pursued moves either along a straight path (linear pursuit) or a circle (circular pursuit). Other than in the non-relativistic case, even the linear pursuit problem is not explicitly integrable. For circular pursuit, the trajectory of the pursuer attains a limit cycle which has the same radius as in the non-relativistic case. Numerical solutions are presented for both cases.