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Prokopenya, Alexander N.

Numerical-symbolic methods for searching relative equilibria in the restricted problem of four bodies. (English) Zbl 1488.70015

Math. Model. Anal. 23, No. 3, 507-525 (2018).
Summary: 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.

Cited in 1 Document

MSC:

70F10 \(n\)-body problems
68W30 Symbolic computation and algebraic computation
70-08 Computational methods for problems pertaining to mechanics of particles and systems

Keywords:

restricted four-body problem; relative equilibria; symbolic-numerical computation

Software:

Mathematica
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Full Text: DOI
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References:

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