## Numerical Solutions of the N-Body ProblemApproach your problem from the right It isn't that they can't see end and begin with the answers. the solution. Then one day, perhaps you will find It is that they can't see the the final question. problem. G.K. Chesterton. The Scandal The Hermit Clad in Crane Feathers in of Father Brown The Point of R. van Gulik's The Chinese Maze Murders. a Pin. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new brancheq. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics, algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisci fI plines as "experimental mathematics", "CFD , "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. |

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### Contents

CONVENTIONAL NUMERICAL METHODS FOR SOLVING | 3 |

THE GENERAL NBODY PROBLEM | 49 |

Method | 94 |

Copyright | |

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### Common terms and phrases

accuracy Algorithm applied asymptotic expansion automatic step Butcher method calculations center of mass coefficients components of velocity compute constants of motion conventional numerical methods convergent coordinates and components denotes discrete initial value discrete mechanics method discretization method equations of motion equilibrium points exact solution explicit method fourth order frame of reference function GO TO LABI Gragg method gravitational constant Greenspan inertial frame infinitely small mass initial value problem integration step interval iteration process Jacobi integral Jacobi's Lemma let us assume Let us note material points method of fourth modified Euler method motion 20 multistep methods N-body problem number of bodies obtain one-step method ordinary differential equations polynomial extrapolation problem of relative procedure proof relative motion Richardson extrapolation rpl rp2 Runge-Kutta method solving the initial step size correction taking into account Theorem total discretization error two-body problem xk+1-xk