Etienne Deprit

Frozen orbits for satellites close to an Earth-like planet

We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ2 throughJ9. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J2 only). In particular, we examine the manner in which the odd zonalJ3 breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vintis problem (J4+J22=0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vintis problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.

Implementing recurrent back-propagation on the connection machine

Abstract The recurrent back-propagation algorithm for neural networks has been implemented on the Connection Machine, a massively parallel processor. Two fundamentally different graph architectures underlying the nets were tested: one based on arcs, the other on nodes. Confirming the predominance of communication over computation, performance measurements underscore the necessity to make connections the basic unit of representation. Comparisons between these graph algorithms lead to important conclusions concerning the parallel implementation of neural nets in both software and hardware.

The Relegation Algorithm

The relegation algorithm extends the method of normalization by Lie transformations. Given a Hamiltonian that is a power series ℋ = ℋ0+ εℋ1+ ... of a small parameter ε, normalization constructs a map which converts the principal part ℋ0into an integral of the transformed system — relegation does the same for an arbitrary function ℋ[G]. If the Lie derivative induced by ℋ[G] is semi-simple, a double recursion produces the generator of the relegating transformation. The relegation algorithm is illustrated with an elementary example borrowed from galactic dynamics; the exercise serves as a standard against which to test software implementations. Relegation is also applied to the more substantial example of a Keplerian system perturbed by radiation pressure emanating from a rotating source.

Runtime Support for Portable Distributed Data Structures

Multipol is a library of distributed data structures designed for irregular applications, including those with asynchronous communication patterns. In this paper, we describe the Multipol runtime layer, which provides an efficient and portable abstraction underlying the data structures. It contains a thread system to express computations with varying degrees of parallelism and to support multiple threads per processor for hiding communication latency. To simplify programming in a multithreaded environment, Multipol threads are small, finite-length computations that are executed atomically. Rather than enforcing a single scheduling policy on threads, users may write their own schedulers or choose one of the schedulers provided by Multipol. The system is designed for distributed memory architectures and performs communication optimizations such as message aggregation to improve efficiency on machines with high communication startup overhead. The runtime system currently runs on the Thinking Machines CM5, Intel Paragon, and IBM SPI, and is being ported to a network of workstations. Multipol applications include an event-driven timing simulator [1], an eigenvalue solver [2], and a program that solves the phylogeny problem [3].

Portable Parallel Irregular Applications

Software developers for distributed memory multiprocessors often complain about the lack of libraries and tools for developing and performance tuning their applications. While some tools exist for regular array-based computations, support for applications with pointer-based data structures, asynchronous communication patterns, or unpredictable computational costs is seriously lacking. In this paper we describe our experience with six irregular applications from CAD, Robotics, Genetics, Physics, and Computer Science, and offer them as application challenges for other systems that support irregular applications. The applications vary in the amount and kind of irregularity. We characterize their irregularity profiles and the implementation problems that arise from those profiles. In addition to performance, one of our goals is to provide implementations that run efficiently with minimal performance tuning across machine platforms, and our designs are influenced by this desire for performance portability. Each of our applications is organized around one or two distributed data structures, which are part of the Multipol data structure library. We describe these data structures, give an overview of some key features in our underlying runtime support, and present performance results for the applications on three platforms.

Paint by number: Uncovering phase flows of an integrable dynamical system

Given an integrable dynamical system with one degree of freedom, ‘‘painting’’ the integral over phase space proves to be a powerful technique for uncovering both global and local behavior.This graphical technique avoids numerical integration, employing instead a nonlinear method of assigning contrasting colors to the energy values to distinguish subtle details of the flow.

Processing Poisson series in parallel

A massively parallel processor proves to be a powerful tool for manipulating large Poisson series. Assuming the series to be sparse, as in most problems of non-linear dynamics, they map into the computer one term per processor. The parallel processor then executes the basic operations in the free algebra of Poisson series over the field of real or rational numbers. We describe a prototype Poisson series processor currently running on the Connection Machine. Timing the calculation of various expansions in the two-body problem demonstrates massive parallelisms potential for tackling the enormous symbolic calculations customary in non-linear dynamics.

Poincaré's méthode nouvelle by skew composition

Poincaré designed the méthode nouvelle in order to build approximate integrals of Hamiltonians developed as series of a small parameter. Due to several critical deficiencies, however, the method has fallen into disuse in favor of techniques based on Lie transformations. The paper shows how to repair these shortcomings in order to give Poincaré’s méthode nouvelle the same functionality as the Lie transformations. This is done notably with two new operations over power series: a skew composition to expand series whose coefficients are themselves series, and a skew reversion to solve implicit vector equations involving power series. These operations generalize both Arbogast’s technique and Lagrange’s inversion formula to the fullest extent possible.