130 research outputs found
Progress in Classical and Quantum Variational Principles
We review the development and practical uses of a generalized Maupertuis
least action principle in classical mechanics, in which the action is varied
under the constraint of fixed mean energy for the trial trajectory. The
original Maupertuis (Euler-Lagrange) principle constrains the energy at every
point along the trajectory. The generalized Maupertuis principle is equivalent
to Hamilton's principle. Reciprocal principles are also derived for both the
generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis
Principle is the classical limit of Schr\"{o}dinger's variational principle of
wave mechanics, and is also very useful to solve practical problems in both
classical and semiclassical mechanics, in complete analogy with the quantum
Rayleigh-Ritz method. Classical, semiclassical and quantum variational
calculations are carried out for a number of systems, and the results are
compared. Pedagogical as well as research problems are used as examples, which
include nonconservative as well as relativistic systems
Investigations on Dynamical Stability in 3D Quadrupole Ion Traps
We firstly discuss classical stability for a dynamical system of two ions levitated in a 3D Radio-Frequency (RF) trap, assimilated with two coupled oscillators. We obtain the solutions of the coupled system of equations that characterizes the associated dynamics. In addition, we supply the modes of oscillation and demonstrate the weak coupling condition is inappropriate in practice, while for collective modes of motion (and strong coupling) only a peak of the mass can be detected. Phase portraits and power spectra are employed to illustrate how the trajectory executes quasiperiodic motion on the surface of torus, namely a Kolmogorov–Arnold–Moser (KAM) torus. In an attempt to better describe dynamical stability of the system, we introduce a model that characterizes dynamical stability and the critical points based on the Hessian matrix approach. The model is then applied to investigate quantum dynamics for many-body systems consisting of identical ions, levitated in 2D and 3D ion traps. Finally, the same model is applied to the case of a combined 3D Quadrupole Ion Trap (QIT) with axial symmetry, for which we obtain the associated Hamilton function. The ion distribution can be described by means of numerical modeling, based on the Hamilton function we assign to the system. The approach we introduce is effective to infer the parameters of distinct types of traps by applying a unitary and coherent method, and especially for identifying equilibrium configurations, of large interest for ion crystals or quantum logic
Cosmological Moduli Dynamics
Low energy effective actions arising from string theory typically contain
many scalar fields, some with a very complicated potential and others with no
potential at all. The evolution of these scalars is of great interest. Their
late time values have a direct impact on low energy observables, while their
early universe dynamics can potentially source inflation or adversely affect
big bang nucleosynthesis. Recently, classical and quantum methods for fixing
the values of these scalars have been introduced. The purpose of this work is
to explore moduli dynamics in light of these stabilization mechanisms. In
particular, we explore a truncated low energy effective action that models the
neighborhood of special points (or more generally loci) in moduli space, such
as conifold points, where extra massless degrees of freedom arise. We find that
the dynamics has a surprisingly rich structure - including the appearance of
chaos - and we find a viable mechanism for trapping some of the moduli.Comment: 35 pages, 14 figures, references adde
Interaction of two charges in a uniform magnetic field
The thesis starts with a short introduction to smooth dynamical systems and
Hamiltonian dynamical systems. The aim of the introductory chapter is to collect basic
results and concepts used in the thesis to make it self–contained.
The second chapter of the thesis deals with the interaction of two charges moving
in R2 in a magnetic field B. This problem can be formulated as a Hamiltonian system
with four degrees of freedom. Assuming that the magnetic field is uniform and the
interaction potential has rotational symmetry we reduce this Hamiltonian system to one
with two degrees of freedom; for certain values of the conserved quantities and choices
of parameters, we obtain an integrable system. Furthermore, when the interaction
potential is of Coulomb type, we prove that, for suitable regime of parameters, there
are invariant subsets on which this system contains a suspension of a subshift of finite
type. This implies non–integrability for this system with a Coulomb type interaction.
Explicit knowledge of the reconstruction map and a dynamical analysis of the reduced
Hamiltonian systems are the tools we use in order to give a description for the various
types of dynamical behaviours in this system: from periodic to quasiperiodic and chaotic
orbits, from bounded to unbounded motion.
In the third chapter of the thesis we study the interaction of two charges moving
in R3 in a magnetic field B. This problem can also be formulated as a Hamiltonian
system, but one with six degrees of freedom. We keep the assumption that the magnetic
field is uniform and the interaction potential has rotational symmetry and reduce this
Hamiltonian system to one with three degrees of freedom; for certain values of the
conserved quantities and choices of parameters, we obtain a system with two degrees
of freedom. Furthermore, when the interaction potential is chosen to be Coulomb we
prove the existence of an invariant submanifold where the system can be reduced by a
further degree of freedom. The reductions simplify the analysis of some properties of
this system: we use the reconstruction map to obtain a classification for the dynamics
in terms of boundedness of the motion and the existence of collisions. Moreover, we
study the scattering map associated with this system in the limit of widely separated
trajectories. In this limit we prove that the norms of the gyroradii of the particles are
conserved during an interaction and that the interaction of the two particles is responsible
for a rotation of the guiding centres around a fixed centre in the case of two charges
whose sum is not zero and a drift of the guiding centres in the case of two charges
whose sum is zero
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