General features of brachistochrones

We discuss new necessary and sufficient conditions for a trajectory to be a brachistochrone—a path of minimum travel time between two points—for a particle subject to a conservative force. We establish these results with two complementary approaches. The first is heuristic, exploiting the fact that any brachistochrone trajectory must be stationary. The second approach involves direct application of the Euler–Lagrange equation. Finally, we show that for a central force there is a conserved quantity with dimension of time, and the wire that guides the particle may be replaced by a suitable magnetic field.