Abstract
The relationship is investigated between a one-dimensional potential and a track in a vertical plane along which a bead is constrained to slide 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 the potential. For a given potential, the isodynamical track is specified in terms of a non-linear first-order ordinary differential equation which depends on the amplitude, and whose relevant solutions may be neither unique nor smooth. Several cases of quadratic and quartic convex functions are solved numerically and displayed. For a given amplitude of oscillation, only the track shape of minimum height is smooth at the origin. The track shapes isodynamical to a double-well (Duffing oscillator) potential for the symmetrical cross-well oscillations are all found to have a kink at the origin. Corresponding to a V-shaped potential, there is a variety of track shapes including one of minimum height which is smooth at the origin.
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