scholarly journals Relative equilibria of a gyrostat in the three body problem

2007 ◽  
Vol 2 ◽  
pp. 475-498
Author(s):  
J. A. Vera ◽  
A. Vigueras
Keyword(s):  
Body Problem ◽  
Relative Equilibria ◽  
Three Body Problem ◽  
Three Body

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 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.

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