Fractional Calculus - Gauge-institute.org

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannonwe look for a differential equation that will describe such motion.4.2 The Tautochrone Differential Equationds( y − η )=−2gdtProof:The potential energymg( y − η)converts into the kinetic energy122⎛ds⎞m ⎜⎜⎝ dt ⎠ ⎟,wheres= s( η)is the arc-length of the curve. Therefore,ds=± 2( gy− η).dtSince the particle speed is in the direction of− y , we havedsdt=− 2( gy−η)s'( η)dη( y − η )=−2gdt.Integrating the right side over the time interval, and the left overthe height interval we obtain an integral equation for the curve ofthe bead’s motion:16