This project deals with three areas of mathematics applied to physical systems or motivated by such applications. The first of these areas is a proposed exploration of a newly discovered connection between the (one-dimensional) Schroedinger's equation and the geometry of "bicycle tracks." Schroedinger's equation is a fundamental model in many areas of mathematics, physics and engineering. Many physical phenomena, such as the spectrum of the hydrogen atom, the working of particle accelerators, and many more are explained by the properties of this equation. A completely different object of geometrical study--the so-called "bicycle tracks"--has been studied for over a century, with no apparent connection to Hill's equations. It was recently noticed by the PI that the two subjects are closely related. Part of proposed research is to exploit this relationship to gain new insights into both subjects. The second area of proposed research deals with systems with rapid imposed vibrations. Such vibrations are used in particle accelerators, in particle traps for low-temperature experiments and in laser "tweezers." Underlying geometry of the phenomenon was understood only recently. The PI proposes to extend his earlier work to broader physical contexts, and to further explore the connection between differential geometry, mechanics and averaging theory in the context of this problem. This work is expected to show how concepts from differential geometry (curvature, normal family) find their manifestations in mechanics. In the third area of project, the researcher's goal is two-fold: first, to develop variational techniques for time--dependent Hamiltonian systems, and second, to shed new light on the problem of Arnold diffusion--an instability problem in Hamiltonian dynamics--in specific examples motivated by physics or geometry.
This project deals with three areas of mathematics arising in physical applications. The first part of the project is an exploration of a newly discovered connection between two fields which until recently seemed unrelated: Schroedinger's equation on the one hand, and the geometry of "bicycle tracks" on the other. The former describes the spectrum of the hydrogen atom, the working of particle accelerators, mechanical vibrations, and more. It was recently noticed by this investigator that the two subjects are closely related. This connection opens exciting prospects of each area giving new insights into the other. One such possibility is a striking connection between the motion of water waves on the one hand, and the deformation of tracks on the other. The second direction of proposed research addresses study of mechanical systems subjected to rapid vibrations. Rapid vibrations have found unexpected practical use in particle accelerators, in particle traps used for low-temperature experiments and in laser "tweezers," the latter enabling biologists to manipulate parts of a cell by light, in an non--invasive way. The gist of a key phenomenon in this area was discovered in the investigator's earlier work. The present project aims to extend this work to wider applications, including fluids and gases subjected to rapid vibration, with applications including cooling, removing impurities from gases, affecting turbulence, and more. This project will apply tools of differential geometry and differential equations to better understand physical phenomena mentioned above. In the third area of project, the researcher's goal is two-fold: first, to develop variational techniques for analyzing time-dependent Hamiltonian systems, and second, to shed new light on the problem of Arnold diffusion--an instability problem in dynamics (e.g., of motion of satellites)--in specific examples motivated by physics and geometry.
The outcomes of this research project include the following. With a graduate student A. Saadatpour, we explained a mechanism by which waves can travel along a chain of coupled oscillators; such chains arise in numerous physical applications. With a graduate student C. Ai, we considered the effects of tidal dissipation in simplified models of celestial mechanics, discovering a counterintuitive fact: a model of three bodies ends in collision or in escape to infinity with probability one. The value of this research is that it promises an extension to more realistic models, and may be applicable to certain more realistic systems. With V. Kaloshing and M. Saprykina we showed that Arnold diffusion happens in a lattice of pendula, a system modeling numerous physical settings. Loosely speaking, Arnold diffusion is the transfer of ``significant" amount of energy between two nodes of a lattice, for arbitrarily weak coupling between nodes. Energy can transfer between the nodes when the two are in resonance. But any such transfer destroys the resonance and thus the transfer; transfers ``self--destruct" after only a small amount of energy is transmitted. It is therefore not at all clear that the ``significant" amount of energy can move from one node to another. We showed that in fact the energy can ``diffuse" between any two nodes. One significance of this finding is that it confirms the belief that Arnold diffusion is a universal phenomenon. The PI discovered that two seemingly different objects: the stationary Schroedinger equation and the pursuit curves ("bike tracks") are equivalent. The first of these two objects is a ubiquitous and fundamental in physics, while the second is appealing geometrically. This observation opens a new perspective of using the extensive theory of Schroedinger equation to find new information on the tracks, with new geometrical interpretations of old results. In addition to these and some other results, the PI published two books, one from Princeton University Press, for general audience of scientifically curious readers: http://press.princeton.edu/titles/9666.html and the other from the AMS, aimed at undergraduate and graduate students in mechanics, calculus of variations and optimal control: www.ams.org/bookstore-getitem/item=stml-69
