This project will develop mathematical tools with the goal of understanding some important physical phenomena, in three different but related areas. The first of these areas deals with the study of mechanical systems subjected to high frequency, high intensity vibrations. Laser tweezers used to manipulate objects inside biological cells belong to this class of systems, as does the Paul trap -? an ion trap for the invention of which W. Paul received the 1989 Nobel Prize in physics. This project aims to develop a new, more intuitive and geometric understanding of these and related phenomena. In addition to explaining physics in a simpler new way, this research would develop new connections between different mathematical areas. The second part of the project deals with Hill's equation, a system of fundamental importance in many areas of mathematics, physics, and engineering. The main aim of this part of the project is to explain a fundamental and elegant feature of this system which has been observed numerically but the reason for which is still not understood. The third part of the project deals with a basic problem of understanding the long-time effect of tidal dissipation on the motion of celestial bodies.
The proposed research consists of three areas unified by the common goal of understanding dynamical systems arising in physical settings. The first of these areas deals with the study of mechanical systems subjected to high frequency, high intensity vibrations. The research will develop a new geometric understanding of related phenomena (as a possible benefit to applications), and will relate averaging theory with differential geometry (as a bridge between two areas of mathematics). The second part of the project will explore the geometry of the bifurcation diagram associated with Hill's equation, to understand qualitatively how the system depends on parameters. The main aim of this part of the project is to explain the collapse of resonance gaps in the Stark effect, which is observed numerically but the reason for which is still not understood. The third part of the project deals with analysis of models for the long-time effect of tidal dissipation on the motion of celestial bodies.
