If the particle's position is parametrized by the arclength s (t) from the lowest point, the kinetic energy is proportional to s ˙ ˙ 2 . The potential energy is proportional to the height y (s). One way the curve can be an isochrone is if the Lagrangian is that of a simple harmonic oscillator : the height of the curve must be proportional to the arclength squared.