In the literature concerned with the relativistic dynamics of a mass-point, the problem of finding geodesics is stated as a variational problem, and the equations for the geodesics are the Euler equations associated with it.
This article discusses the case of geodesics of null-length in the Riemannian space-time of General Relativity, as they play an important role, in particular in connection with the hypothetic motion of photons in a gravitational field. It is shown that this problem is not in general a variational problem, but a problem of controllability.
Let the system under consideration be a masspoint with (constant) proper mass m0 > 0, and consider the one-player differential game (i.e. the problem of closed-loop control) defined by (a) the set & X T of states y = (x,t) of the system, where & is a domain of a 3-dimensional Euclidean space R3, containing state (O,O), and T=(--oo,+co); (b) the set U of strategies u( .), which are functions defined on & X T + , where T + = (0, + oo), of class C' on & and piece-wise continuous on T + , such that Kqvwordr. Controllability, Differential qualitative games, Con-u(x,t) EK(x,t) c R3 trol theory, Relativistic dynamics, Geodesics.