EXACT MIMICRY OF NONLINEAR OSCILLATORY POTENTIAL MOTION - NONUNIQUENESS OF ISODYNAMICAL TRACKS

The relationship is investigated between a one-dimensional potential a nd a track in a vertical plane along which a bead is constrained to sl ide freely under the influence of gravity, such that the motion of the bead, projected onto the horizontal axis, is exactly the same as (i.e ., is isodynamical to) the one-dimensional oscillatory motion due to t he potential. For a given potential, the isodynamical track is specifi ed in terms of a non-linear first-order ordinary differential equation which depends on the amplitude, and whose relevant solutions may be n either unique nor smooth. Several cases of quadratic and quartic conve x functions are solved numerically and displayed. For a given amplitud e of oscillation, only the track shape of minimum height is smooth at the origin. The track shapes isodynamical to a double-well (Duffing os cillator) potential for the symmetrical cross-well oscillations are al l found to have a kink at the origin. Corresponding to a V-shaped pote ntial, there is a variety of track shapes including one of minimum hei ght which is smooth at the origin. (C) 1997 Academic Press Limited.