Symplectic Integration of Hamiltonian Systems using Polynomial Maps

In order to perform numerical studies of long-term stability in nonlinear Hamiltonian systems, one needs a numerical integration algorithm which is symplectic. Further, this algorithm should be fast and accurate. In this paper, we propose such a symplectic integration algorithm using polynomial map refactorization of the symplectic map representing the Hamiltonian system. This method should be particularly useful in long-term stability studies of particle storage rings in accelerators.Comment: 10 pages, 1 figur

On the symplectic structures arising in Optics

Geometric optics is analysed using the techniques of Presymplectic Geometry. We obtain the symplectic structure of the space of light rays in a medium of a non constant refractive index by reduction from a presymplectic structure, and using adapted coordinates, we find Darboux coordinates. The theory is illustrated with some examples and we point out some simple physical applicationsComment: AmsTeX file and 2 figures (epsf required). To appear in Forsch. der Physik. This version replaces that of (96/02/09) where postcript files containing figures were corrupte

Accurate Transfer Maps for Realistic Beamline Elements: Part I, Straight Elements

The behavior of orbits in charged-particle beam transport systems, including both linear and circular accelerators as well as final focus sections and spectrometers, can depend sensitively on nonlinear fringe-field and high-order-multipole effects in the various beam-line elements. The inclusion of these effects requires a detailed and realistic model of the interior and fringe fields, including their high spatial derivatives. A collection of surface fitting methods has been developed for extracting this information accurately from 3-dimensional field data on a grid, as provided by various 3-dimensional finite-element field codes. Based on these realistic field models, Lie or other methods may be used to compute accurate design orbits and accurate transfer maps about these orbits. Part I of this work presents a treatment of straight-axis magnetic elements, while Part II will treat bending dipoles with large sagitta. An exactly-soluble but numerically challenging model field is used to provide a rigorous collection of performance benchmarks.Comment: Accepted to PRST-AB. Changes: minor figure modifications, reference added, typos corrected

How Wigner Functions Transform Under Symplectic Maps

It is shown that, while Wigner and Liouville functions transform in an identical way under linear symplectic maps, in general they do not transform identically for nonlinear symplectic maps. Instead there are ``quantum corrections'' whose hbar tending to zero limit may be very complicated. Examples of the behavior of Wigner functions in this limit are given in order to examine to what extent the corresponding Liouville densities are recovered.Comment: 8 pages, 6 figures [RevTeX/epsfig, macro included]. To appear in Proceedings of the Advanced Beam Dynamics Workshop on Quantum Aspects of Beam Physics (Monterey, CA 1998

A Christian Response to the Restrictions of Girls\u27 Education in Afghanistan Under the Taliban Regime: How Kuyperian Insight Requires Theological and Embodied Engagement

Jaelyn Dragt, a Dordt University junior, majoring in Social Work and Community Development and minoring in Theology, submitted this essay to the Lambertus Verburg Prize for Excellence in Kuyperian Scholarship competition, 2023

Beautiful Risk: A New Psychology of Loving and Being Loved (Book Review)

Reviewed Title: The Beautiful Risk: A New Psychology of Loving and Being Loved, by James Olthuis. Grand Rapids: Zondervan, 2001

Exact evolution of time-reversible symplectic integrators and their phase error for the harmonic oscillator

The evolution of any factorized time-reversible symplectic integrators, when applied to the harmonic oscillator, can be exactly solved in a closed form. The resulting modified Hamiltonians demonstrate the convergence of the Lie series expansions. They are also less distorted than modified Hamiltonian of non-reversible algorithms. The analytical form for the modified angular frequency can be used to assess the phase error of any time-reversible algorithm.Comment: Submitted to Phys. Lett. A, Six Pages two Column

Factoring the unitary evolution operator and quantifying entanglement

The unitary evolution can be represented by a finite product of exponential operators. It leads to a perturbative expression of the density operator of a close system. Based on the perturbative expression scheme, we present a entanglement measure, this measure has the advantage that it is easy to compute for a general dynamical process.Comment: 11 pages, LATEX, no figure