| Equilibrium problems for elastic rods: Equilibrium configurations of elastic rods under various constraints have been studied since the time of Euler and Kirchhoff. Our focus, with an eye to DNA (see below), has been on the effect of intrinsic curvature and/or flexibility, using a perturbation analysis to understand the symmetry-breaking that occurs when these imperfections are first introduced into the rod. We have also worked to develop efficient algorithms for numerically determining equilibria using the parameter continuation package AUTO. |
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| Elastic rod models for biological macromolecules: We are investigating to what extent elastic rod models can be used to model biological polymers such as DNA or proteins. Many have used rods to understand DNA; our focus has been on incorporating DNA intrinsic shape and flexibility, to model the cyclization experiments of Jason Kahn (University of Maryland College Park). A new focus is to apply these techniques to coiled-coil proteins, as part of a nanotechnology project with biology, chemistry, and physics faculty at Haverford. |
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Stability of Equilibria in the Calculus of Variations:
Elastic rod equilibrium problems are part of a
family of infinite-dimensional optimization problems in a field
called the calculus of variations. To understand the
physical relevance of these equilibria, we must determine their stability,
i.e., distinguish local minima from saddle points.
In classic calculus of variations theory,
this is done via conjugate points. We have worked to
extend the idea of conjugate points to include isoperimetric
constraints and integral second variation operators.
This extended stability theory was then applied to:
* rods with constant curvature and elliptical cross-section held in a multi-covered ring * intrinsically straight elliptical rods buckled under twist and endloading * intrinsically straight elliptical rods buckled under endloading and an external field modeling a "soft" wall |
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| Semiclassical approximations in quantum physics: In my doctoral research with Greg Ezra at Cornell, we studied quantum mechanical systems that began to take on classical-like qualities (say, for the energy sufficiently high). In that limit, one can approximate the quantum solution with appropriately combined classical solutions, with what are often called semiclassical sum-over-classical-trajectories formulas. We looked at these approximations in small chemical systems and studied how the underlying classical dynamics affects the efficient computation of these formulas. We focused on systems with singular potentials, and used a quantum regularization in conjunction with semiclassical approximations of the quantum path integral to produce a more accurate approximation to the quantum mechanics. You can see an example of the effect of the regularization for a 1 degree-of-freedom singular potential. |
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