Solution of the Dynamic Lame Problem

The well-known Lame problem, posed in 1852, involves solving the static equilibrium of a parallelepiped with free side surfaces subjected to action of opposite end forces. In this article, the same problem for a more complicated case of impacts of end forces is considered.

An exact analytical solution of this problem is found.

Emphasizing the particular difficulty of solving this problem, Lamé, in his book “Leçons sur la thorie mathematique de Ielasticite des corps solides” (Paris, 1852), wrote: “C’est une sorte d’engine aussi digne d’exercer la sagasite des analystes que le fameux problem des trios corps de la Mécanique celeste”,—“This is a kind of drive, as worthy of training the clairvoyance of analysts as the famous three-body problem of celestial mechanics.” At that time, this problem was the subject of a prize from the Paris Academy of Sciences, that was intended for the one who solved the Lamé problem. Despite this, to date, no solution has been found even for a static case of this problem, not to mention the complicated version of the problem.