Share:


Numerical-symbolic methods for searching relative equilibria in the restricted problem of four bodies

Abstract

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.

Keyword : restricted four-body problem, relative equilibria, symbolic-numerical computation, Wolfram Mathematica

How to Cite
Prokopenya, A. (2018). Numerical-symbolic methods for searching relative equilibria in the restricted problem of four bodies. Mathematical Modelling and Analysis, 23(3), 507-525. https://doi.org/10.3846/mma.2018.030
Published in Issue
Jul 4, 2018
Abstract Views
70
PDF Downloads
66
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1] A. Albouy, H.E. Cabral and A.A. Santos. Some problems of the classical n-body problem. Celestial Mechanics and Dynamical Astronomy, 113(4):369–375, 2012. https://doi.org/10.1007/s10569-012-9431-1.

[2] M. Alvares-Ramirez, J.E.F. Skea and T.J. Stuchi. Nonlinear stability analysis in a equilateral restricted four-body problem. Astrophysics and Space Science, 358(1):3, 2015. https://doi.org/10.1007/s10509-015-2333-4.

[3] R.E. Arenstrof. Central configurations of four bodies with one inferior mass. Celestial Mechanics, 28(1-2):9–15, 1982. https://doi.org/10.1007/s10509-015-2333-4.

[4] D. Boccaletti and G. Pucacco. Theory of orbits. Volume 1: Integrable systems and non-perturbative methods. 3rd edn. Astronomy and Astrophysics Library. Springer-Verlag, Berlin Heidelberg, 2004. https://doi.org/10.1007/s10509-015-2333-4.

[5] V.A. Brumberg. Celestial mechanics: past, present, future. Solar System Research, 47(5):347–358, 2013. https://doi.org/10.1134/S0038094613040011.

[6] D.A. Budzko and A.N. Prokopenya. Symbolic-numerical analysis of equilibrium solutions in a restricted four-body problem. Programming and Computer Software, 36(2):68–74, 2010. https://doi.org/10.1134/S0038094613040011.

[7] D.A. Budzko and A.N. Prokopenya. On the stability of equilibrium positions in the circular restricted four-body problem. In V.P. Gerdt, W. Koepf, E.W. Mayr and E.V. Vorozhtsov(Eds.), Computer Algebra in Scientific Computing, volume 6885 of Lecture Notes in Computer Science, pp. 88–100, Berlin, Heidelberg, 2011. Springer-Verlag. https://doi.org/10.1007/978-3-642-23568-9 8.

[8] D.A. Budzko and A.N. Prokopenya. Stability of equilibrium positions in the spatial circular restricted four-body problem. In V.P. Gerdt, W. Koepf, E.W. Mayr and E.V. Vorozhtsov(Eds.), Computer Algebra in Scientific Computing, volume 7442 of Lecture Notes in Computer Science, pp. 72–83, Berlin, Heidelberg, 2012. Springer-Verlag. https://doi.org/10.1007/978-3-642-32973-9 7.

[9] D.A. Budzko and A.N. Prokopenya. Symbolic-numerical methods for searching equilibrium states in a restricted four-body problem. Programming and Computer Software, 39(2):74–80, 2013. https://doi.org/10.1134/S0361768813020035.

[10] E.A. Grebenikov, E.V. Ikhsanov and A.N. Prokopenya. Numeric-symbolic computations in the study of central configurations in the planar Newtonian fourbody problem. In V.G. Ganzha, E.W. Mayr and E.V. Vorozhtsov(Eds.), Computer Algebra in Scientific Computing, volume 4194 of Lecture Notes in Computer Science, pp. 192–204, Berlin, Heidelberg, 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1134/S0361768813020035.

[11] R. Kozera. Asymptotics for length and trajectory from cumulative chord piecewise-quarties. Fundamenta Informaticae, 61(3):267–283, 2004.

[12] A.P. Markeev. Libration points in celestial mechanics and cosmic dynamics. Nauka, Moscow, 1978. (in Russian)

[13] J.I. Palmore. Collinear relative equilibria of the planar n-body problem. Celestial Mechanics, 28(1):17–24, 1982. https://doi.org/10.1134/S0361768813020035.

[14] A.N. Prokopenya. Computing the stability boundaries for the Lagrange triangular solutions in the elliptic restricted three-body problem. Mathematical Modelling and Analysis, 11(1):95–104, 2006. https://doi.org/10.1080/13926292.2006.9637305.

[15] A.N. Prokopenya. Hamiltonian normalization in the restricted many-body problem by computer algebra methods. Programming and Computer Software, 38(3):156–166, 2012. https://doi.org/10.1134/S0361768812030048.

[16] A.N. Prokopenya. Symbolic-numerical analysis of the relative equilibria stability in the planar circular restricted four-body problem. In V.P. Gerdt, W. Koepf, W.M. Seiler and E.V. Vorozhtsov(Eds.), Computer Algebra in Scientific Computing, volume 10490 of Lecture Notes in Computer Science, pp. 329–345, Cham, 2017. Springer International Publishing. https://doi.org/10.1007/978- 3-319-66320-3 24.

[17] A.E. Roy. Orbital motion. 4th edn. Institute of Physics Publishing, Bristol and Philadephia, 2005.

[18] D.G. Saari. On the role and the properties of n-body central configurations. Celestial Mechanics, 21:9–20, 1980. https://doi.org/10.1007/BF01230241.

[19] C. Sim´o. Relative equilibrium solutions in the four-body problem. Celestial Mechanics, 18(2):165–184, 1978. https://doi.org/10.1007/BF01228714.

[20] S. Smale. Mathematical problems for the next century. The mathematical Intelligencer, 20(2):7–15, 1998. https://doi.org/10.1007/BF03025291.

[21] K.F. Sundman. M´emoire sur le probl`eme des trios corps. Acta Mathematica, 36(1):105–179, 1912.

[22] V.G. Szebehely. Theory of orbits. The restricted problem of three bodies. Academic Press, New York, London, 1967. Translated from Astronomicheskii Zhurnal, vol. 46, no. 2, pp. 459–460

[23] A. Wintner. The analytical foundations of Celestial Mechanics. Princeton University Press, Princeton, New York, 1941.

[24] S. Wolfram. An elementary introduction to the Wolfram Language. 2nd edition. Wolfram Media, Champaign, IL, USA, 2017.