Stability and Instability in Mechanical Systems

Background in Hamiltonian systems and averaging methods
The geometric description of mechanical systems is Hamiltonian mechanics. This geometric description allows to make systematically changes of variables which make the system look simple. Depending on the details of the simplified model one may obtain stability or some geometric structures that lead to instability.

Amadeu Delshams, Universitat Politècnica de Catalunya
Anatoly Neishtadt, Moscow Space Research Institute-Loughborough University

Classical and global variational methods
For a long time it has been recognized that the trajectories of physical systems are critical points of some action. For a long time, variational methods have served to construct solutions with some global properties or to join together several short term solutions. Starting in the early 80's, variational methods in mechanics have flourished by incorporating new concepts and techniques and drawing connections to other branches of mathematics such as viscosity solutions.

Chong-Qing Cheng, University of Nanjing
Massimiliano Berti, Università di Napoli

Invariant objects: KAM theorem and Normally hyperbolic invariant manifolds
Invariant objects are landmarks that organize the long term behavior of systems. Depending on their relative geometry, they may act as barriers, hence prove stability, or as routes to scape. Quasiperiodic solutions (invariant tori) and normally hyperbolic invariant manifolds are the most systematically studied invariant objects besides periodic orbits.

Chris Jones , University of North Carolina
Rafael de la Llave, University at Austin, Texas

Shadowing lemmas and topological methods
The shadowing lemmas conclude (under appropriate hypothesis) that given an approximate orbit there is a true orbit nearby. They are a basic tool to join the information obtained from the study of invariant objects or to validate approximate (e.g. numerical) computations.

Ernest Fontich, Universitat de Barcelona
Marian Gidea, Northeastern Illinois University

Homoclinic connections and separatrix map
Orbits that oscillate between to invariant objects are the main mechanism for chaotic behavior and the basis of many mechanisms for instability.

Rafael Ramírez-Ros, Universitat Politècnica de Catalunya
Dmitry Treshev, Moscow State University

Applications to Celestial Mechanics and Chemistry
The subject of Mechanics was created to explain the motion of celestial bodies. In more modern times it was realized that important parts of chemistry and reaction dynamics can be explained by the motion of atoms interacting under well understood laws. The recent advances in Mechanics can throw light on some of the observed phenomena. Many observed phenomena still lack explanation and provide stimulus for new progress in Mechanics.

Carles Simó, Universitat de Barcelona
Turgay Uzer, Georgia Institute of Technology