Relativistic tautochrone Available to Purchase

The path joining two points A and B, which a particle falling from rest in a uniform gravitational field must adopt, so that the time of transit from A to B is independent of the location of A is called the tautochrone. In the nonrelativistic case, the path is known to be a cycloid, a standard method associated with the derivation of this result being, for example, the method of Laplace transforms. For the relativistic case that is studied herein, the methods of fractional calculus are shown to be more useful in the derivation of the exact relativistic tautochrone. This latter derivation is then checked out by the Laplace transform approach for the relativistic problem, from the point of view of consistency. Using the same method, the relativistic tautochrone associated with a charged particle of charge q and mass m falling from rest in an uniform electric field is also worked out. The tautochrone turns out to be an incomplete elliptic function of the second kind E(δ,r). As an application, the power radiated by the charged particle as it accelerates along the curve is then computed; it is found to be proportional to (1 − v²/c²)⁻², with v(c) the velocity of the particle (light). Finally, an appendix highlights the utility of fractional calculus vis‐á‐vis the approach of Abel for the relativistic tautochrone.