PHYS 6102- Classical Mechanics II  (Spring 2013)

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 Wednesday April 17 A. Kamor Annular billiard dynamics in a Hamiltonian Framework We analyze the dynamics of a valence electron of the buckminsterfullerene molecule (C60) subjected to a circularly polarized laser field by modeling it with the motion of a classical particle in an annular billiard. We show that the phase space of the billiard model gives rise to three distinct trajectories: Whispering gallery orbits'', which only hit the outer billiard wall, daisy orbits'' which hit both billiard walls (while rotating solely clockwise or counterclockwise for all time), and orbits which only visit the downfield part of the billiard, as measured relative to the laser term. These trajectories, in general, maintain their distinct features, even as intensity is increased from 10^10 to 10^14 W*cm-2. We attribute this robust separation of phase space to the existence of twistless tori. Friday April 19 M. Dimitriyev Review of Planar Vortices: Integrability, Geometric Phase, and Statistics The study of vorticity has proven to be an indispensable part of fluid dynamics: in viscous fluids it provides an avenue for dissipation and in nonviscous fluids it is a conserved quantity. I will first present a brief history of planar vortices in nonviscous fluids and their mathematical treatment. I will then review a proof of the integrability of the planar three-vortex problem and discuss the non-integrability of systems with more vortices. Finally, I will discuss the presence of a geometric phase that occurs in the three-vortex setting and address the possibility of negative temperature in the statistical mechanics of vortices. Monday April 22 M. Jon De Viana Aubry Mather Theory via the Frenkel-Kontorova model A model for one-dimensional crystals, proposed by Frenkel and Kontorova, leads to the study of the following situation: Consider a (possibly infinite) chain of identical atoms which interact only with their nearest neighbor, in the same way as two nodes connected by a spring. The atoms are also acted upon by an external force which is periodic. A crystal in this model is given by a stable configuration, and this means that any $\ell^1$- perturbation of the positions of the atoms of the chain increases the total energy (maybe in the talk we will simplify this a little and just consider perturbations of finitely many atoms). The objective of the talk is to present a theory due to Serge Aubry (and later refined by Victor Bangert) that permits to study the existence and structure of the set of minimal configurations. If time allows, we will comment how this theory is useful for other seemingly unrelated problems. Wednesday April 24 M. Kingsbury Terradynamics in Granular Media Dynamical theory models for owing media of air and water predict animal movement through these materials through the use of lift, drag, and thrust forces. Models of terrestrial locomotion, however, are focused on solid ground interactions and do not account for granular media where the substrate may flow in response to intrusion. The article in focus develops a force model for intruding bodies moving freely in granular media to predict a small legged robot's locomotion on this material over different leg shapes and kinematics. The study discovered that stresses in the granular media were dependent upon the depth, orientation, and direction of motion of the intruder which in turn affects the locomotion of the robot. In particular this project will examine and present this granular resistive force theory model for generalized intruders into the granular media. Through understanding the ground reactive forces and yields of the material, insight into the resultant locomotion of a body can be extended. Friday April 26 H. Tossounian Boltzmann's H-Theorem and Kac's Master Equation First, we introduce Boltzmann's problem for space homogenous particles, define the entropy of a distribution, and prove Boltzmann's H-theorem: that H(f(t; ~v)) increases. All this is done in McKeans's paper. Next we show how a solution to the linear Kac Master equation having the "Boltzmann property" can lead a solution to Boltzmann's problem.