Homogenization in time of singularly perturbed mechanical systems / Folkmar Bornemann

... The purpose of this monograph is to present a particular method for an explicit homogenization of certain singularly perturbed mechanical systems. Caused by singular perturbations and properties of the model, the solutions to these systems will show up rapid microscale fluctuations. The method is based on energy principles and weak convergence techniques. Since nonlinear functionals are not weakly sequentially continuous, simultaneously the weak limits of all those nonlinear quantities of the rapidly oscillating components which are of importance for the underlying problem are studied. Using physically motivated concepts of virial theorems, adiabatic invariants, and resonances, sufficiently many relations between all these weak limits are established, allowing to calculate them explicitly.

In the introductory chapter all prerequisites about weak convergence are introduced which are needed in the first three chapters of the monograph.

In Chapter 2, mechanical systems are studied on Riemannian manifolds, singulary perturbed by a strong constraining potential.

A homogenization result is stated and proved. As a particular subcase, the microscale justification of the Lagrange-d’Alembert principle is discussed by utilizing strong constraining potentials. Giving unified proofs for the results known on this justification, the necessity is shown of the conditions which prior to this work were only known to be sufficient.

Chapter 3 continues with several applications. Thus the problem of guiding center motion in plasma physics and an elimination of fast vibrations in molecular dynamics are discussed. Finally, an application consists of a simplified, finite-dimensional version of a model in quantum chemistry which describes a coupling of a quantum mechanical system with a classical one. It is shown that this model can be transformed to the case of Chapter 2.

The corresponding infinite-dimensional coupling model, in its original, untransformed guise leads to the second major case of this monograph, subject of Chapter 4. (Zentralblatt)