scholarly journals NUMERICAL-SYMBOLIC METHODS FOR SEARCHING RELATIVE EQUILIBRIA IN THE RESTRICTED PROBLEM OF FOUR BODIES

2018 ◽  
Vol 23 (3) ◽  
pp. 507-525 ◽  
Author(s):  
Alexander Prokopenya
Keyword(s):  
Body Problem ◽  
Relative Equilibrium ◽  
Numerical Procedure ◽  
Relative Equilibria ◽  
Algebraic Equations ◽  
Three Body Problem ◽  
Starting Point ◽  
Equilibrium Configurations ◽  
Four Body Problem ◽  
Three Body

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.

scholarly journals Properties of the lunar gravity assisted transfers from LEO to the retrograde-GEO

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bo-yong He ◽  
Peng-bin Ma ◽  
Heng-nian Li
Keyword(s):  
Body Problem ◽  
Space Debris ◽  
Low Earth Orbit ◽  
Earth Orbit ◽  
Restricted Three Body Problem ◽  
The Sun ◽  
Lunar Gravity ◽  
Three Body Problem ◽  
Four Body Problem ◽  
Three Body

AbstractThe retrograde geostationary earth orbit (retro-GEO) is an Earth’s orbit. It has almost the same orbital altitude with that of a GEO, but an inclination of 180°. A retro-GEO monitor-satellite gives the GEO-assets vicinity space-debris warnings per 12 h. For various reasons, the westward launch direction is not compatible or economical. Thereby the transfer from a low earth orbit (LEO) to the retro-GEO via once lunar swing-by is a priority. The monitor-satellite departures from LEO and inserts into the retro-GEO both using only one tangential maneuver, in this paper, its transfer’s property is investigated. The existence of this transfer is verified firstly in the planar circular restricted three-body problem (CR3BP) model based on the Poincaré-section methodology. Then, the two-impulse values and the perilune altitudes are computed with different transfer durations in the planar CR3BP. Their dispersions are compared with different Sun azimuths in the planar bi-circular restricted four-body problem (BR4BP) model. Besides, the transfer’s inclination changeable capacity via lunar swing-by and the Sun-perturbed inclination changeable capacity are investigated. The results show that the two-impulse fuel-optimal transfer has the duration of 1.76 TU (i.e., 7.65 days) with the minimum values of 4.251 km s−1 in planar CR3BP, this value has a range of 4.249–4.252 km s−1 due to different Sun azimuths in planar BR4BP. Its perilune altitude changes from 552.6 to 621.9 km. In the spatial CR3BP, if the transfer duration is more than or equal to 4.00 TU (i.e., 17.59 days), the lunar gravity assisted transfer could insert the retro-GEO with any inclination. In the spatial BR4BP, the Sun’s perturbation does not affect this conclusion in most cases.

Lagrangian relative equilibria for a gyrostat in the three-body problem

Open Physics ◽  
2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Juan López
Keyword(s):  
Body Problem ◽  
Sufficient Conditions ◽  
Relative Equilibria ◽  
Rigid Bodies ◽  
Necessary And Sufficient Conditions ◽  
Three Body Problem ◽  
Legendre Series ◽  
Three Body ◽  
Necessary And Sufficient ◽  
Number Of Equilibria

AbstractIn this paper we consider the noncanonical Hamiltonian dynamics of a gyrostat in the three-body problem. By means of geometric mechanics methods, we study the approximate Poisson dynamics that arise when we develop the potential of the system in Legendre series and truncate this to an arbitrary order k. After reduction of the dynamics by means of the two symmetries of the system, we consider the existence and number of equilibria which we denominate of Lagrangian type, in analogy with classic results on the topic. Necessary and sufficient conditions are established for their existence in an approximate dynamics of order k, and explicit expressions for these equilibria are given, this being useful for the subsequent study of their stability. The number of Lagrangian equilibria is thoroughly studied in approximate dynamics of orders zero and one. The main result of this work indicates that the number of Lagrangian equilibria in an approximate dynamics of order k for k ≥1 is independent of the order of truncation of the potential, if the gyrostat S 0 is almost spherical. In relation to the stability of these equilibria, necessary and sufficient conditions are given for linear stability of Lagrangian equilibria when the gyrostat is almost spherical. In this way, we generalize the classical results on equilibria of the three-body problem and many results provided by other authors using more classical techniques for the case of rigid bodies.

scholarly journals Three-body problem - from Newton to supercomputer plus machine learning

2021 ◽  
Author(s):  
Shijun LIAO ◽  
Xiaoming Li ◽  
Yu Yang
Keyword(s):  
Machine Learning ◽  
Periodic Orbits ◽  
High Performance ◽  
Body Problem ◽  
Ann Model ◽  
Three Body Problem ◽  
Starting Point ◽  
Families Of Periodic Orbits ◽  
Artificial Neural Network Ann ◽  
Three Body

Abstract The famous three-body problem can be traced back to Newton in 1687, but quite few families of periodic orbits were found in 300 years thereafter. In this paper, we propose an effective approach and a roadmap to numerically gain planar periodic orbits of three-body systems with arbitrary masses by means of machine learning based on an artificial neural network (ANN) model. Given any a known periodic orbit as a starting point, this approach can provide more and more periodic orbits (of the same family name) with variable masses, while the mass domain having periodic orbits becomes larger and larger, and the ANN model becomes wiser and wiser. Finally, we have an ANN model trained by means of all obtained periodic orbits of the same family, which provides a convenient way to give accurate enough predictions of periodic orbits with arbitrary masses for physicists and astronomers. It suggests that the high-performance computer and artificial intelligence (including machine learning) should be the key to gain periodic orbits of the famous three-body problem.

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