# Many Body Symmetrical Dynamical Systems

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)

### Abstract

In this thesis we investigate the stability of few body symmetrical dynamical systems which include four and five body symmetrical dynamical systems. We divide this thesis into three parts.

In the first part we determine some special analytical solutions for restricted, coplanar, four body problem both with equal masses (Roy and Steves, 1998) and with two pairs of equal masses. The Lagrange solutions L1 and L4 are obtained in the two triangular solutions. The equilateral triangle and the isosceles triangle solutions are also obtained. But this is not enough to know the existence of such solutions; therefore we discuss their linear stability by applying a small perturbation to the position of the equilibrium point. All the equilibrium solutions both equal mass (Gomatan, Steves and Roy, 1999) and with two pairs of equal masses are proved to be linearly unstable. We also provide a comprehensive literature review on the four and five body problems to put our research on these problems in the wider context.

In the second part we investigate more complicated and general four body problems. We analyze numerically the stability of the phase space of the Caledonian Symmetric Four Body Problem (CSFBP), a symmetrically restricted four body configuration first introduced by Roy and Steves (1998), by perturbing the position of one of the bodies and using the general four body equations. We show that the CSFBP is stable towards small perturbations and there is no significant change in the symmetry before and after the perturbations.

In the third part we introduce a stationary mass to the centre of mass of the CSFBP, to derive analytical stability criterion for this five body system and to use it to discover the effect on the stability of the whole system by adding a central body. To do so we define a five body system in a similar fashion to the CSFBP which we call the Caledonian Symmetric Five body Problem (CS5BP). We determine the maximum value of the Szebehely constant, Co = 0.659, for which the CS5BP system is hierarchically stable for all mass ratios. The CS5BP system has direct applications in Celestial Mechanics. The analytical stability criterion tells us for what value of Co the system will be hierarchically stable but we would like to know what happens before this point. To understand this, we determine a numerical stability criterion for the CS5BP system which compares well with the analytical stability criterion derived earlier. We conclude this thesis with the generalization of the above analytical stability criterion to the n body symmetrical systems. This new system which we call the Caledonian Symmetric N Body Problem (CSNBP) has direct applications in Celestial Mechanics and Galactic dynamics.

Research presented in this thesis includes the following original investigations: determination of some analytical solutions of the four body problem and their linear stability analysis; stability analysis of the near symmetric coplanar CSFBP ; derivation of the analytical stability criterion valid for all time for a special symmetric configuration of the general five-body problem, the CS5BP, which exhibits many of the salient characteristics of the general five body problem; numerical investigation of the hierarchical stability of the CS5BP and derivation of the stability criterion for the CSNBP.
Date of Award 2004 English Glasgow Caledonian University Bonnie Steves (Supervisor)

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