Variational Integrators for Almost-Integrable Systems

We construct several variational integrators--integrators based on a discrete variational principle--for systems with Lagrangians of the form L = L_A + epsilon L_B, with epsilon << 1, where L_A describes an integrable system. These integrators exploit that epsilon << 1 to increase their accuracy by constructing discrete Lagrangians based on the assumption that the integrator trajectory is close to that of the integrable system. Several of the integrators we present are equivalent to well-known symplectic integrators for the equivalent perturbed Hamiltonian systems, but their construction and error analysis is significantly simpler in the variational framework. One novel method we present, involving a weighted time-averaging of the perturbing terms, removes all errors from the integration at O(epsilon). This last method is implicit, and involves evaluating a potentially expensive time-integral, but for some systems and some error tolerances it can significantly outperform traditional simulation methods.Comment: 14 pages, 4 figures. Version 2: added informative example; as accepted by Celestial Mechanics and Dynamical Astronom

An Overview of the 13:8 Mean Motion Resonance between Venus and Earth

It is known since the seminal study of Laskar (1989) that the inner planetary system is chaotic with respect to its orbits and even escapes are not impossible, although in time scales of billions of years. The aim of this investigation is to locate the orbits of Venus and Earth in phase space, respectively to see how close their orbits are to chaotic motion which would lead to unstable orbits for the inner planets on much shorter time scales. Therefore we did numerical experiments in different dynamical models with different initial conditions -- on one hand the couple Venus-Earth was set close to different mean motion resonances (MMR), and on the other hand Venus' orbital eccentricity (or inclination) was set to values as large as e = 0.36 (i = 40deg). The couple Venus-Earth is almost exactly in the 13:8 mean motion resonance. The stronger acting 8:5 MMR inside, and the 5:3 MMR outside the 13:8 resonance are within a small shift in the Earth's semimajor axis (only 1.5 percent). Especially Mercury is strongly affected by relatively small changes in eccentricity and/or inclination of Venus in these resonances. Even escapes for the innermost planet are possible which may happen quite rapidly.Comment: 14 pages, 11 figures, submitted to CMD

A pilgrimage to gravity on GPUs

In this short review we present the developments over the last 5 decades that have led to the use of Graphics Processing Units (GPUs) for astrophysical simulations. Since the introduction of NVIDIA's Compute Unified Device Architecture (CUDA) in 2007 the GPU has become a valuable tool for N-body simulations and is so popular these days that almost all papers about high precision N-body simulations use methods that are accelerated by GPUs. With the GPU hardware becoming more advanced and being used for more advanced algorithms like gravitational tree-codes we see a bright future for GPU like hardware in computational astrophysics.Comment: To appear in: European Physical Journal "Special Topics" : "Computer Simulations on Graphics Processing Units" . 18 pages, 8 figure

Environment-Induced Decoherence and the Transition From Quantum to Classical

We study dynamics of quantum open systems, paying special attention to those aspects of their evolution which are relevant to the transition from quantum to classical. We begin with a discussion of the conditional dynamics of simple systems. The resulting models are straightforward but suffice to illustrate basic physical ideas behind quantum measurements and decoherence. To discuss decoherence and environment-induced superselection einselection in a more general setting, we sketch perturbative as well as exact derivations of several master equations valid for various systems. Using these equations we study einselection employing the general strategy of the predictability sieve. Assumptions that are usually made in the discussion of decoherence are critically reexamined along with the ``standard lore'' to which they lead. Restoration of quantum-classical correspondence in systems that are classically chaotic is discussed. The dynamical second law -it is shown- can be traced to the same phenomena that allow for the restoration of the correspondence principle in decohering chaotic systems (where it is otherwise lost on a very short time-scale). Quantum error correction is discussed as an example of an anti-decoherence strategy. Implications of decoherence and einselection for the interpretation of quantum theory are briefly pointed out.Comment: 80 pages, 7 figures included, Lectures given by both authors at the 72nd Les Houches Summer School on "Coherent Matter Waves", July-August 199

Proposal to ARPA for Research on Intelligent Automata and Micro-Automation

The results of a decade of work in Artificial Intelligence have brought us to the threshold of a new phase of knowledge-based programming -- in which we can design computer systems that (1) react reasonably to significantly complicated situations and (2) perhaps more important for the future -- interact intelligently with their operators when they encounter limitations, bugs or insufficient information. This proposal lays out programmes for bringing several such systems near to the point of useful application. These include: A physical "micro-automation" system for maintenance and repair of electronic circuits. A related "expert" problem-solving program for diagnosis and modification of electronic circuits. A set of advanced "Automatic Programming" techniques and systems for aid in developing and debugging large computer programs. Some Advanced Natural Language application methods and sustems for use with these and other interactive projects. A series of specific "expert" problem solvers, including Chess analysis. Steps toward a new generation of more intelligent Information Retrieval and Management Assistance systems