A Zero - Dimensional Shock.

Abstract

In this note, we add a new wrinkle to the very old problem of determining the motion of a mass point on a spring. We adopt a general model for the spring in which the force needed to compress it to zero length is infinite. (Consequently the motion is governed by a singular nonlinear second-order ordinary differential equation.) In this setting we entertain the possibility, permitted by the governing equations, that such a total compression is actually attained. This total compression corresponds to a kind of shock. We then extract from the governing equations all the illumination they can shed on the physical behavior. Our problem, which presents novel features for ordinary differential equations, captures in microcosm deep and unresolved issues involving shocks and their suppression, which arise in the study of quasilinear hyperbolic and parabolic partial differential equations. We comment briefly on these issues in Section 8.