Numeric-Symbolic Computations in the Study of Central Configurations in the Planar Newtonian Four-Body Problem

The symmetric central configurations of the 4-body problem with masses m1=m2≠mÁlvarez-Ramírez1, Llibre2 2013Applied Mathematics and Computation14070View full textAdd to dashboardCite

“…We must mention that the results of these two theorems are compatible with the numerical results obtained by Simó in [26], and by Grebenikov, Ikhsanov and Prokopenya [9].…”

“…Now we consider the resultant h(a) of the polynomials g 1 (a, b) with g 2 (a, b) with respect to the variable b. Therefore h(a) is a polynomial in the variable a with coefficients polynomials in the variable m. More precisely, h(a) is 268435456m 12 (1 + m) 4 a 4 (1 + 4a 2 ) 9 (4a + 4a 2 + m 2 ) 4 (−4032a 6 + 768a 8 + 3072a 10 + 4096a 12 + 16ma 3 + 192ma 5 + 8192ma 6 + 768ma 7 + 1024ma 9…”

Numerical-Symbolic Methods for Searching Relative Equilibria in the Restricted Problem of Four Bodies

“…In such a system the particles P 1 , P 2 are immovable and located on the same straight line which may be considered as the Ox axis. Without loss of generality, one can consider also that the dimensionless x-coordinates of particles P 1 , P 2 are equal to 1 and a, respectively, where the variable a is defined by the equation (see [10,17,22]…”

“…In the framework of the restricted three-body problem it is assumed that negligibly small mass of particle P 3 (µ 3 = m 3 /m 0 = 0) does not affect the motion of massive particles P 0 , P 1 , P 2 , and so equations (2.1) and (2.2) determining equilibrium positions of particles P 2 and P 3 are solved separately. However, if mass of particle P 3 is taken into account (µ 3 > 0) the system of equations determining equilibrium configurations of the particles becomes more complicated and may be written in the form (see [10]) where…”

“…, L 5 in case of m 2 = 0. In this way we obtain the restricted four-body problem that has been a subject of many papers (see, for example, [2,6,7,8,9,10,11,13,19]). It should be emphasized that in the framework of this model, motion of the massive particles P 0 , P 1 , P 2 is given and is determined by the corresponding solutions of the three-body problem, namely, by the Lagrange triangular solutions or by the Euler collinear solutions (see [17]).…”