Variational Aspects Of The Slow Mechanical System

This chapter describes the variational property of the slow mechanical system. The main goal is to derive some properties of the “channel” and information about the Aubrey sets. More precisely, the chapter proves Proposition 5.1. It provides a condition for the “width” of the channel to be non-zero. The chapter then discusses the limit of the set, which corresponds to the “bottom” of the channel. It drops all subscripts “s” to simplify the notations. The results proved in the chapter are mostly contained in John Mather's works. The chapter reformulates some of them for its purpose and also provides some different proofs.

A variational property of the von Kármán plate problem

The solution of the boundary value problem of anisotropic Föppl–von Kármán plates is shown to be a critical point for a suitable energy functional. Moreover, under the assumption that the minimum of the total energy exists, we prove a saddle-point property and also deduce from it the form of the boundary conditions for plates clamped on part of the boundary and loaded on the complementary part.