Paul E. Nacozy

The Use of Integrals in Numerical Integrations of the N-Body Problem

The numerical integration of systems of differential equations that possess integrals is often approached by using the integrals to reduce the number of degrees of freedom or by using the integrals as a partial check on the resulting solution, retaining the original number of degrees of freedom.Another use of the integrals is presented here. If the integrals have not been used to reduce the system, the solution of a numerical integration may be constrained to remain on the integral surfaces by a method that applies corrections to the solution at each integration step. The corrections are determined by using linearized forms of the integrals in a least-squares procedure.The results of an application of the method to numerical integrations of a gravitational system of 25-bodies are given. It is shown that by using the method to satisfy exactly the integrals of energy, angular momentum, and center of mass, a solution is obtained that is more accurate while using less time of calculation than if the integrals are not satisfied exactly. The relative accuracy is ascertained by forward and backward integrations of both the corrected and uncorrected solutions and by comparison with more accurate integrations using reduced step-sizes.

The intermediate anomaly

AbstractA Sundman time transformation of the form

Matrix formulation of the Picard method for parallel computation

dt = cr^n ds

A discussion of long-term numerical solutions of the Jupiter-Saturn-Sun system

, wheren=3/2 andc is a constant, is integrated analytically for Keplerian motion. The integral involves an elliptic integral of the first kind. The variable s is called theintermediate anomaly.

Time elements in Keplerian orbital elements

The geopotential expansion is givenentirely in terms of nonsingular orbital elements. The expansion and its derivatives are valid for zero eccentricity and inclination. The development begins with the geopotential expansion in singular, classical elements as given by Izsak (1964), Allan (1965) and Kaula (1966). The singular geopotential is then transformed into a nonsingular set of elements

A DISCUSSION OF THE SOLUTION FOR THE MOTION OF PLUTO

The increasing availability of computing machines capable of parallel computation has accelerated interest in numerical methods that exhibit natural parallel structures. In particular, the parallel structure of the Picard method of successive approximations for the numerical solution of ordinary differential equations allows straightforward adaptation of the method for use on parallel computers. A matrix formulation of the Picard method for parallel computation is presented here in which the numerical solution is obtained in truncated Chebyshev series. The application of the formulation to parallel processing computing machines is discussed.

On the long-term motion of Pluto

Hansen method of partial anomalies for cometary orbits applied to comet Encke perturbed by earth and compared with perturbations by numerical integration

A discussion of time transformations and local truncation errors

A detailed discussion of recent numerical studies concerning the stability of the Jupiter-Saturn-Sun system is presented.