Nonlinear Dynamics with Applications to Physical Systems

Project: Research project

Project Details

Description

Proposal ID: 0605878 PI: Levi, Mark Insitution: Pennsylvania State Univ University Park Title: Nonlinear Dynamics with Applications to Physical Systems Proposed research consists of three different parts, united by the common theme of understanding dynamical systems arising in physical applications. In part I, a geometrical approach to the problem of parametric resonance aims at gaining new insight into the model that arises in numerous applications in mechanics, physics, engineering, and plays an important role inmathematics. The problem of parametric resonance has been extensively studied by analysts, but recent numerical observations suggest that a topological approach may give a principally new insight. This problem is treated in many texts in mathematics, physics and engineering; basic new insight into the problem will be of considerable interest. It is hoped that this research will shed new light on Stark effect (splitting of atomic spectral lines). Part II of proposed research addresses study of systems with imposed rapid vibrations. Stabilization by vibration is used in particle acclerators, in particle traps and in laser ``tweezers". Underlying geometry of the phenomenon was understood only recently. The author proposes to extend his earlier work to broader physical contexts, and to further explore the fruitful connection between differential geometry, mechanics and averaging theory. This work will show how concepts from differential geometry (curvature, normal family) find their manifestations in mechanics. Part III of proposed research deals with Arnold diffusion — a fundamental aspect of stability of Hamiltonian systems. 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 in specific examples motivated by physics or geometry. The unifying theme of proposed research is to establish new connections between abstract mathematical concepts on the one hand and their physical manifestations (e.g., in mechanics) on the other. Such connections enrich mathematics and benefit applications by providing the latter with tools for better understanding physical phenomena. A recent example of such mutually beneficial interaction was the author's use of differential geometry to provide new insight into the functioning of the Paul trap — a device used to suspend charged particles by electric field. In 1989 W. Paul was awarded Nobel prize for his invention. Proposed research should give new insights into some resonance phenomena of basic importance in mechanics, quantum mechanics and engineering. It is hoped that some results of proposed research will make their way into upper undergraduate and graduate texts in differential equations, engineering and mechanics. Both graduate and undergraduate students will be closely involved with this research.

StatusFinished
Effective start/end date6/1/065/31/11

Funding

  • National Science Foundation: $276,923.00
  • National Science Foundation: $276,923.00

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