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

### From NZJM

**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

mailto:giovanni.mingari@unibo.it

**Daniele Ritelli**

Dipartimento di matematica

per le scienze economiche e sociali

viale Filopanti, 5

40127 Bologna

ITALY

mailto:daniele.ritelli@unibo.it

**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:
is met, where are fixed and *A* and *k*
depend on 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.