A brief introduction to symplectic integrators and recent results

The author begins with a brief synopsis about Hamiltonian systems and symplectic maps. A symplectic integrator is a symplectic map {phi}(q,p;t) that systematically approximates the time t flow of a Hamiltonian system. Systematic means: (1) in time step, t, i.e. the error should vanish as some power of the time step, and (2) in order of approximation, i.e. one would like a hierarchy of such {phi} that have errors that vanish as successively higher powers of the time step. At present the authors known two general types of symplectic integrators: (1) implicit integrators that are derived from a generating function or from algebraic conditions on Runge-Kutta schemes, and (2) explicit integrators that are derived from integrable Hamiltonians or from algebraic conditions on Runge-Kutta schemes

Two-Stream Instability Model With Electrons Trapped in Quadrupoles

We formulate the theory of the two-stream instability (e-cloud instability) with electrons trapped in quadrupole magnets. We show that a linear instability theory can be sensibly formulated and analyzed. The growth rates are considerably smaller than the linear growth rates for the two-stream instability in drift spaces and are close to those actually observed

Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions

We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the phase error can also be eliminated two orders higher than that of the integrator. The use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth accuracy in the periodic energy. We study the 1-D harmonic oscillator and the 2-D Kepler problem in great details, and compare the effectiveness of some recent fourth order algorithms.Comment: Submitted to Phys. Rev. E, 29 Page

Symplectic integrators with adaptive time steps

In recent decades, there have been many attempts to construct symplectic integrators with variable time steps, with rather disappointing results. In this paper we identify the causes for this lack of performance, and find that they fall into two categories. In the first, the time step is considered a function of time alone, \Delta=\Delta(t). In this case, backwards error analysis shows that while the algorithms remain symplectic, parametric instabilities arise because of resonance between oscillations of \Delta(t) and the orbital motion. In the second category the time step is a function of phase space variables \Delta=\Delta(q,p). In this case, the system of equations to be solved is analyzed by introducing a new time variable \tau with dt=\Delta(q,p) d\tau. The transformed equations are no longer in Hamiltonian form, and thus are not guaranteed to be stable even when integrated using a method which is symplectic for constant \Delta. We analyze two methods for integrating the transformed equations which do, however, preserve the structure of the original equations. The first is an extended phase space method, which has been successfully used in previous studies of adaptive time step symplectic integrators. The second, novel, method is based on a non-canonical mixed-variable generating function. Numerical trials for both of these methods show good results, without parametric instabilities or spurious growth or damping. It is then shown how to adapt the time step to an error estimate found by backward error analysis, in order to optimize the time-stepping scheme. Numerical results are obtained using this formulation and compared with other time-stepping schemes for the extended phase space symplectic method.Comment: 23 pages, 9 figures, submitted to Plasma Phys. Control. Fusio

Explicit Lie-Poisson integration and the Euler equations

We give a wide class of Lie-Poisson systems for which explicit, Lie-Poisson integrators, preserving all Casimirs, can be constructed. The integrators are extremely simple. Examples are the rigid body, a moment truncation, and a new, fast algorithm for the sine-bracket truncation of the 2D Euler equations.Comment: 7 pages, compile with AMSTEX; 2 figures available from autho

Unequal relationships in high and low power distance societies: a comparative study of tutor - student role relations in Britain and China

This study investigated people's conceptions of an unequal role relationship in two different types of society: a high power distance society and a low power distance society. The study focuses on the role relationship of tutor and student. British and Chinese tutors and postgraduate students completed a questionnaire that probed their conceptions of degrees of power differential and social distance/closeness in this role relationship. ANOVA results yielded a significant nationality effect for both aspects. Chinese respondents judged the relationship to be closer and to have a greater power differential than did British respondents. Written comments on the questionnaire and interviews with 9 Chinese academics who had experienced both British and Chinese academic environments supported the statistical findings and indicated that there are fundamental ideological differences associated with the differing conceptions. The results are discussed in relation to Western and Asian concepts of leadership and differing perspectives on the compatibility/incompatibility of power and distance/closeness

An Exactly Conservative Integrator for the n-Body Problem

The two-dimensional n-body problem of classical mechanics is a non-integrable Hamiltonian system for n > 2. Traditional numerical integration algorithms, which are polynomials in the time step, typically lead to systematic drifts in the computed value of the total energy and angular momentum. Even symplectic integration schemes exactly conserve only an approximate Hamiltonian. We present an algorithm that conserves the true Hamiltonian and the total angular momentum to machine precision. It is derived by applying conventional discretizations in a new space obtained by transformation of the dependent variables. We develop the method first for the restricted circular three-body problem, then for the general two-dimensional three-body problem, and finally for the planar n-body problem. Jacobi coordinates are used to reduce the two-dimensional n-body problem to an (n-1)-body problem that incorporates the constant linear momentum and center of mass constraints. For a four-body choreography, we find that a larger time step can be used with our conservative algorithm than with symplectic and conventional integrators.Comment: 17 pages, 3 figures; to appear in J. Phys. A.: Math. Ge

Climate variability and ice-sheet dynamics during the last three glaciations

AbstractA composite North Atlantic record from DSDP Site 609 and IODP Site U1308 spans the past 300,000 years and shows that variability within the penultimate glaciation differed substantially from that of the surrounding two glaciations. Hematite-stained grains exhibit similar repetitive down-core variations within the Marine Isotope Stage (MIS) 8 and 4–2 intervals, but little cyclic variability within the MIS 6 section. There is also no petrologic evidence, in terms of detrital carbonate-rich (Heinrich) layers, for surging of the Laurentide Ice Sheet through the Hudson Strait during MIS 6. Rather, very high background concentration of iceberg-rafted debris (IRD) indicates near continuous glacial meltwater input that likely increased thermohaline disruption sensitivity to relatively weak forcing events, such as expanded sea ice over deepwater formation sites. Altered (sub)tropical precipitation patterns and Antarctic warming during high orbital precession and low 65°N summer insolation appear related to high abundance of Icelandic glass shards and southward sea ice expansion. Differing European and North American ice sheet configurations, perhaps aided by larger variations in eccentricity leading to cooler summers, may have contributed to the relative stability of the Laurentide Ice Sheet in the Hudson Strait region during MIS 6

A consideration of the challenges involved in supervising international masters students

This paper explores the challenges facing supervisors of international postgraduate students at the dissertation stage of the masters programme. The central problems of time pressure, language difficulties, a lack of critical analysis and a prevalence of personal problems among international students are discussed. This paper makes recommendations for the improvement of language and critical thinking skills, and questions the future policy of language requirements at HE for international Masters students