ISODYNAMICAL TRACKS AND POTENTIALS

The correspondence is found between a track in a vertical plane along which a bead is constrained to slide freely under the influence of gra vity, and a one-dimensional potential, such that the motion due to the potential is exactly the same as (i.e., is isodynamical to) the motio n of the bead, projected onto the horizontal axis. For any given track shape, the shape of the isodynamical one-dimensional potential functi on is explicitly and uniquely specified, and in general depends on the amplitude of oscillation. Various examples for quadratic and quartic functions are solved and displayed. In particular, the potential isody namical to sufficiently large oscillations on a double valley shaped t rack has a triple well form. Other isodynamical situations dealt with include the case in which the non-linear potential and track functions are the same, and the case of motion along the are length of the trac k path itself rather than its projection. The special cases of V-shape d and W-shaped tracks/potentials are solved in an Appendix. (C) 1997 A cademic Press Limited.