PLANAR BRACHISTOCHRONE OF A PARTICLE ATTRACTED IN VACUO BY AN INFINITE ROD

The authors analyze the planar brachistochrone in vacuo under the attraction of an infinite rod, adding a new closed form treatment to the known solutions’ collection. Accordingly, a nonlinear boundary value problem is solved after solution’s existence and uniqueness are proved noticing that the variational integrand meets the conditions of a Cesari’s theorem. This problem, proposed by G. T. Tee in "Brachistochrones for attractive logarithmic potential" New Zealand Journal of Mathematics 30, No.2, 183–196 (2001) and treated numerically, is solved here in closed form. The trajectory’s parametric equations are obtained by means of a generalized, 2-variables, hypergeometric Lauricella confluent function, for the first time used in optimization.