The General Problem of the Stability of Motion

Abstract

The book consists of 28 papers, presented at the NATO Advanced Study Institute on the Fundamentals of Friction, held in July 1991 in the German Harz mountains. The objective of the Institute was to bring together experts from the two fields of "classical" macroscopic surface mechanics, and "modern" surface science, including molecular dynamics and point contact microscopy of atomic resolution. Thus, friction phenomena are approached by macroscopic models and experiments, as well as by what is being learned at the microscopic level (in current jargon referred to as nano tribology). The papers are divided into seven chapters. Five deal with the classical macro aspects, while, not unexpectedly, only one is devoted to the atomic scale approach. One final chapter brings together three papers on self-induced stick-slip vibrations and instabilities in friction systems. The five macromechanical papers are mainly based on K. L. Johnson's continuum mechanical formulation of surface mechanics. (Johnson contributes an Appendix, summarizing the concepts of his well-known book "Contact Mechanics.") After an initial chapter with broader perspectives on energy dissipation (by D. Tabor) and fluctuations in friction (by E. Rabinowicz), a second chapter is focused on adhesion contributions to frictional forces, both between smooth and rough surfaces and between particles in granular materials. Chapter 3 deals with fracture, deformation, and shear of the sliding interfaces, while Chapters 4 and 5 treat lubricated contacts from the tribochemical point of view and with respect to molecularly thin surface films. The sixth chapter represents the atomic scale approach to friction. An introductory paper by McClelland and Glosli presents two atomistic models for friction as well as frictionless and wearless sliding contacts. The models are applied in molecular dynamics calculations of frictional force and energy and are compared to the results of atomic force microscope (AFM) measurements. In a second paper a team of researchers from the Institute of Physics at the University of Basel gives an update on Friction Force Microscopy (FFM). The authors also report experiments on dry mica and magneto optical disks, and of 2 and 4-molecular-layer soap films. A final group of three papers deal with computational and simulation techniques for atomic scale surface contacts. Ferrante and Bozzolo from NASA Lewis Research Center in Cleveland discuss the use of different potential functions (first principles compared BOOKS RECEIVED

Sufficient conditions for robustness of $$\mathcal{K}\mathcal{L}$$ -stability for difference inclusions

“…In continuous-time, results are available as far back as Lyapunov's original work [13] where it was proved that a Lyapunov function exists for linear differential equations with an asymptotically stable origin. Further work generated Lyapunov functions for nonlinear differential equations with an asymptotically stable origin [9,14] or asymptotically stable closed sets [3,20].…”

On non-linear flows with slip boundary condition

“…Unlike perturbation techniques [25][26][27][28][29], the HAM is independent of small/large parameters. Unlike all other reported perturbation and non-perturbation techniques such as the artificial small parameter method [30], the δ−expansion method [31,32] and decomposition method [33,34], the HAM provides us with a simple way to adjust and control the convergence region and rate of approximation series. The HAM has already been successfully applied to many non-linear problems [35][36][37][38][39][40][41][42][43][44][45][46][47][48].…”

On integrability of a heavy rigid body sinking in an ideal fluid

“…For quasihomogeneous equations the analysis of solution branching can be done using the Lyapunov-Yoshida method [23], [29]. The idea of the method is that under a certain variable change the equations of motion take the form Proof.…”

“…The block corresponding to equations (8) has eigenvalues 4 and 1/2 ± √ 17/2 . As the latter are irrationals, we conclude after [23] and [29] that the general solution of equations (3)-(5) branches on the complex time plane. As the solution is not meromorphic, we conclude after [17], [1] that equations (3)-(5) are not algebraically integrable.…”