Bulletin of the American Physical Society
2014 Annual Meeting of the Mid-Atlantic Section of the APS
Volume 59, Number 9
Friday–Sunday, October 3–5, 2014; University Park, Pennsylvania
Session E2: Theoretical Physics |
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Chair: Michael Widom, Carnegie-Mellon University Room: Life Sciences Building 004 |
Saturday, October 4, 2014 3:30PM - 3:42PM |
E2.00001: Supersymmetry and Lie Algebras Patrick Moylan Construction of quotients, or localization as it is called in mathematics, provides a powerful tool to relate different physical structures which share some underlying similarities. We recall the Gelfand-Kirillov conjecture which assert that the quotient field of the universal enveloping algebra of a Lie algebra is isomorphic to some skew field extension of a Weyl algebra. Another example is the isomorphism between certain skew field extensions of the universal enveloping algebras of the Poincare Lie algebra and the Lie algebra of the de Sitter group established by P. Bozek, M. Havlicek and O. Navratil (cf. M. Havlicek, P. Moylan (1993), J. Math. Phys., 34 (11) 5320-5332). Working in a quantum groups setting we extend these ideas to include supersymmetry. We show how it is possible to describe q deformations of superalgebras in terms enveloping algebras of Lie algebras by localizations and extensions of them. At least for certain low rank simple superalgebras our results lead to new representations. In particular, we obtain new representations of the standard q deformation of the orthosymplectic Lie super algebra osp(1\textbar 2). References: 1) P. Moylan (2014) J. Phys. Conf. Ser., (512) 012026; 2) S. Clark, W. Wang (2013) Lett. Math. Phys. (103) 207-231. [Preview Abstract] |
Saturday, October 4, 2014 3:42PM - 3:54PM |
E2.00002: Equivalence Between Geodesic and Lagrangian Formulations of Pendulum Systems on Manifolds Jared Bland The pendulum system may be described in terms of the Lagrangian related to the system. This Lagrangian is defined on a manifold (higher-dimensional surface) in the configuration space, which is determined from the constraints of the system. Alternatively, if a proper manifold is chosen, then the shortest paths, or geodesics, will trace out the motion of the system. This provides another method to formulate mechanics. Rather than creating constraints as in the Lagrangian approach, we find these geodesics. Both methods are rooted in solving variation problems. A basic case is presented: A two-pendulum system without gravity or coupling. Since there is no gravity the angular velocity is constant, and the answer intuitive, but the difference between the two methods may be illuminated. We also suggest related open problems, such as incorporating a surface gravity of $g$ and coupling between the pendulums. Coupled pendula are well studied using Lagrangian mechanics for small oscillations, but not without the small oscillation assumption. We hope that the method may extend to provide insight into this and similar problems. [Preview Abstract] |
Saturday, October 4, 2014 3:54PM - 4:06PM |
E2.00003: Towards a Conceptual Model of Quantum Mechanics Carl Frederick With the decline of the Copenhagen interpretation of quantum mechanics and the recent experiments indicating that quantum mechanics does actually embody 'objective reality', we propose a a 'mechanical', conceptual model for quantum mechanics. We note that space-time vacuum energy fluctuations imply curvature fluctuations. And those fluctuations are indicated by fluctuations of the metric tensor. The metric tensor fluctuations can 'explain' the uncertainty relations and non-commuting properties of conjugate variables. It also argues that the probability density $\Psi \ast \quad \Psi $ is proportional to the square root of minus the determinant of the metric tensor (the differential volume element) \textbar $-$ g$\mu \nu $\textbar . We further argue that the metric elements are actually not stochastic but are torsionally oscillating at a sufficiently high frequency that measured values of same \textit{appear}stochastic. This is required to allow that the position probability density be a \textit{non}-stochastic variable. An oscillating metric yields, among other things, a model of superposition, photon polarization, and entanglement, and all within the confines of a 4-dimensional space-time. The proposed model is one of 'objective reality' but, of course, as required by Bell's theorem, at the expense of temporal locality. [Preview Abstract] |
Saturday, October 4, 2014 4:06PM - 4:18PM |
E2.00004: A Singularity Handling Approach for the Rayleigh-Plesset Equation Asish Balu, Michael Kinzel, Scott Miller Cavitation dynamics of a nuclei are largely governed by the Rayleigh-Plesset Equation. The research focuses on improving the numerical efficiency of integrating the Rayleigh-Plesset equation with the use of ``singularity handling'' to enable stable integration at much larger time steps, which greatly reduces the computational time while maintaining solution accuracy. In this paper, various singularity-handling algorithms are explored and assessed, where a ``triangle Runge-Kutta backtrace method'' was found to be most effective. In order to maintain constant time step size while maintaining solution quality, the Rayleigh Plesset equation is solved reverse to ensure the solution recovers symmetry across the collapse event. The results indicate that an error of 7{\%} can be maintained while performing over 980{\%} faster than the conventional constant time step Euler method. In addition, the backtrace method had the lowest percent deviation from the actual solution (-0.22{\%}). This increase in efficiency and accuracy allows the program to provide useful solutions in the field of fluids engineering, particularly in the study of shock tube gas explosions. [Preview Abstract] |
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