Symbolic-Numerical Analysis of the Relative Equilibria Stability in the Planar Circular Restricted Four-Body Problem

Prokopenya, Alexander N.

doi:10.1007/978-3-319-66320-3_24

Cited by 2 publications

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“…Given µ 1 > 0 and µ 2 ≥ 0, one can easily find each of these roots numerically with the aid of the built-in Mathematica function F indRoot, for example (see [24]). Equilibrium position of the particle P 3 of negligible mass corresponding to some given configuration of the particles P 0 , P 1 , P 2 is determined by the system of two equations (see [16])…”

Section: General Analysis Of Equilibrium Configurationsmentioning

confidence: 99%

“…Therefore, given the values of µ 1 , µ 2 we have to look for both equilibrium configuration of the massive particles P 0 , P 1 , P 2 and equilibrium positions of the particle P 3 as solutions of the corresponding nonlinear algebraic equations. Only afterwards we can analyze the Hamiltonian function in the neighborhood of each equilibrium configuration and investigate the stability of equilibrium positions (see [14,15,16]).…”

Section: Introductionmentioning

confidence: 99%

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

Prokopenya

2018

Mathematical Modelling and Analysis

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We discuss here the problem of solving the system of two nonlinear algebraic equations determining the relative equilibrium positions in the planar circular restricted four-body problem formulated on the basis of the Euler collinear solution of the three-body problem. The system contains two parameters $\mu_1$, $\mu_2$ and all its solutions coincide with the corresponding solutions in the three-body problem if one of the parameters equals to zero. For small values of one parameter the solutions are found in the form of power series in terms of this parameter, and they are used for separation of different solutions and choosing the starting point in the numerical procedure for the search of equilibria. Combining symbolic and numerical computation, we found all the equilibrium positions and proved that there are 18 different equilibrium configurations of the system for any reasonable values of the two system parameters $\mu_1$, $\mu_2$. All relevant symbolic and numerical calculations are performed with the aid of the computer algebra system Wolfram Mathematica.

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Section: General Analysis Of Equilibrium Configurationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%