QYMSYM: A GPU-Accelerated Hybrid Symplectic Integrator That Permits Close Encounters

We describe a parallel hybrid symplectic integrator for planetary system integration that runs on a graphics processing unit (GPU). The integrator identifies close approaches between particles and switches from symplectic to Hermite algorithms for particles that require higher resolution integrations. The integrator is approximately as accurate as other hybrid symplectic integrators but is GPU accelerated.Comment: 17 pages, 2 figure

Discrete Routh Reduction

This paper develops the theory of abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry. The reduction of variational Runge-Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical $J_2$ correction, as well as the double spherical pendulum. The $J_2$ problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a nontrivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the noncanonical nature of the symplectic structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added, fixed typo

How parents choose to use CAM: a systematic review of theoretical models

Background: Complementary and Alternative Medicine (CAM) is widely used throughout the UK and the Western world. CAM is commonly used for children and the decision-making process to use CAM is affected by numerous factors. Most research on CAM use lacks a theoretical framework and is largely based on bivariate statistics. The aim of this review was to identify a conceptual model which could be used to explain the decision-making process in parental choice of CAM. Methods: A systematic search of the literature was carried out. A two-stage selection process with predetermined inclusion/exclusion criteria identified studies using a theoretical framework depicting the interaction of psychological factors involved in the CAM decision process. Papers were critically appraised and findings summarised. Results: Twenty two studies using a theoretical model to predict CAM use were included in the final review; only one examined child use. Seven different models were identified. The most commonly used and successful model was Andersen's Sociobehavioural Model (SBM). Two papers proposed modifications to the SBM for CAM use. Six qualitative studies developed their own model. Conclusion: The SBM modified for CAM use, which incorporates both psychological and pragmatic determinants, was identified as the best conceptual model of CAM use. This model provides a valuable framework for future research, and could be used to explain child CAM use. An understanding of the decision making process is crucial in promoting shared decision making between healthcare practitioners and parents and could inform service delivery, guidance and policy