Generalizations of the Brachistochrone Problem
  Consider a frictionless surface S in a gravitational field that need not be uniform. Given two points A and B on S, what curve is traced out by a particle that starts at A and reaches B in the shortest time? This paper considers this problem on simple surfaces such as surfaces of revolution and solves the problem two ways: First, we use conservation of mechanical energy and the Euler-Lagange equation; second, we use geometrical optics and the eikonal equation. We conclude with a discussion of the relativistic effects at relativistic velocities. 
J. Gemmer (1 and 2), R. Umble (2), M. Nolan (2) ((1) University of   Arizona, (2) Millersville University)