# Browsing by Subject "Variational principle"

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Item Rapid frequency chirps of an Alfvén wave in a toroidal plasma(2013-05) Wang, Ge, active 2013; Berk, H. L.Show more Results from models that describe frequency chirps of toroidal Alfvén eigenmode excited by energetic particles are presented here. This structure forms in TAE gap and may or may not chirp into the continuum. Initial work described the particle wave interaction in terms of a generic Hamiltonian for the particle wave interaction, whose spatial dependence was xed in time. In addition, we have developed an improved adiabatic TAE model that takes into account the spatial prole variation of the mode and the nite orbit excursion from the resonant ux surfaces, for a wide range of toroidal mode numbers. We have shown for the generic xed prole model that the results from the adiabatic model agree very well with simulation result except when the adiabatic condition breaks down due to the rapid variations of the wave amplitude and chirping frequency. We have been able to solve the adiabatic problem in the case when the spatial prole is allowed to vary in time, in accord with the structure of the response functions, as a function of frequency. All the models predict that up-chirping holes do not penetrate into the continuum. On the other hand clump structures, which down chirp in frequency may, depending on detailed parameters, penetrate the continuum. The systematic theory is more restrictive than the generic theory, for the conditions that enable clump to penetrate into the continuum. In addition, the systematic theory predicts an important nite drift orbit width eect, which eventually limits and suppresses a down-chirping response in the lower continuum. This interruption of the chirping occurs when the trapped particles make a transition from intersecting both resonant points of the continuum to just one resonant point.Show more Item Temporal insights from the end of space(2017-08-24) Feng, Justin Christopher; Matzner, Richard A. (Richard Alfred), 1942-; Gleeson, Austin M; Morrison, Philip J; Hazeltine, Richard D; Gompf, Robert EShow more This dissertation concerns the Weiss variation, its application in both classical and quantum General Relativity, and the role of spatial boundary conditions in characterizing time evolution. I review the Weiss variation in mechanics and classical field theory, and present a novel geometric derivation of the Weiss variation for the gravitational action: the Einstein-Hilbert action plus the Gibbons-Hawking-York boundary term. In particular, I use the first and second variation of area formulas (I include a derivation accessible to physicists in an appendix) to interpret and vary the Gibbons-Hawking-York boundary term. Though the Weiss variation of the gravitational action is in principle known to the relativity community, the variation of area approach formalizes the derivation, and facilitates the discussion of time evolution in General Relativity. I demonstrate the utility of the Weiss variation in quantum General Relativity by presenting a formal derivation of the Wheeler-DeWitt equation from the functional integral of quantum General Relativity by way of boundary variations. One feature of this approach is that it does not require an explicit 3+1 splitting of spacetime in the bulk. For spacetimes with spatial boundary, I show that variations in the induced metric at the spatial boundary can be used to describe time evolution--time evolution in quantum General Relativity is therefore governed by boundary conditions on the gravitational field at the spatial boundary.Show more