Ball Rolling without Slipping inside a Vertical Cylinder

"Suppose a sphere is started rolling on the inside of a rough vertical cylinder (gravity acting, but no dissipative forces); what happens? The only sensible guess is a spiral descent of increasing steepness; actually the sphere moves up and down between two fixed horizontal planes. Golfers are not so unlucky as they think." As quoted in J. E. Littlewood, "From the Mathematical Tripos," A Mathematician's Miscellany, London: Methuen and Company Limited, 1953 pp. 44–47.