The Iterative Solutions of the Analytical N-Body Problem

Recent Advances in Astrodynamics 1960

“…Three methods of qualitative investigations are described by Leimanis including dimensional analysis, continuous induction and invariant measures. Rauch (216) utilized "Steffensen coefficients'' that were extended to the n-body problem by Cheney, for the iterative solution of the n-body problem in real time. A third rather sophisticated mathematical formulation of the n-body problem is carried out by Grobner (87) who utilizes the LIE series and reproduces the known algebraic integrals.…”

Contribution to the solution of the three‐body problem in power series form

“…He succeeded, in fact, in representing the solution of this particular problem by convergent power series. STEFFENSEN'S ideas have subsequently been generalized by several investigators, notably by RAUCH [2] and RIDDEL [3], to the n-body (n > 3) problem using as the independent variable either the time or a regularizing variable. Numerical computations using STEFFENSEN'S procedure have been limited, however, to special investigations of the planar restricted problem in which a rotating reference frame has been adopted.…”

“…Since the variable x must also be changed cyclically into y and z, there is a system of nine differential equations of the second order of type (2) interlaced by the masses and the mutual distances of the three bodies.…”

“…We now expand the solution of ( 2 ) in the form of a MACLAURIN series as follows where I t has been demonstrated by SUNDMAN [15] that if t = o is a non-collision point, that is, if none of the initial distances is vanishing, then the power series of type (6) is convergent in the neighborhood of t = 0. Taking xio and xil equal to the initial conditions (I), these power series represent the unique solution of the problem which satisfies the given initial conditions.…”

“…For v = 2 , the expression for xi2 is simply given by I/Z [d2xi/dt2]r=o, an expression which can be obtained by directly evaluating the right-hand side of equation ( 2 ) a t z = 0 . For v > 2 , the expression for xi, can be derived by repeated differentiation of equation (2). According to LEIBNITZ'S rule we obtain where for the sake of simplicity the symbolic operator D has been used in place of d/dz.…”