# Planar brachistochrone of a particle attracted in vacuo by an infinite rod

### New Zealand Journal of Mathematics

Vol. 39, (2009), Pages 67-77

Giovanni Mingari Scarpello

Dipartimento di matematica

per le scienze economiche e sociali

viale Filopanti, 5

40127 Bologna

ITALY

Daniele Ritelli

Dipartimento di matematica

per le scienze economiche e sociali

viale Filopanti, 5

40127 Bologna

ITALY

Abstract 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: $\begin{cases}y^{\prime }(x) \; = \; -\sqrt{\frac{A^{2}}{\ln \left( ky_{1}\right) -\ln y}-1},\\ y(x_{1})=y_{1},\\y(x_{2})=y_{2},\end{cases}$ is met, where $x_{1},\,x_{2},\,y_{1},\,y_{2},$ are fixed and A and k depend on $x_{i},\,y_{i}$ and on the initial speed. The 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.J.~Tee ('Brachistochrones for attractive logarithmic potential'), [21], 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.

Keywords Brachistochrone, Nonlinear boundary value problem, Lauricella hypergeometric functions.

Classification (MSC2000) Primary: 49K15; Secondary: 33D60.