Computational studies in the planar symmetric six-body problem : this dissertation is submitted for the degree of Doctor of Philosophy, School of Mathematical and Computational Sciences, Massey University

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Massey University
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This thesis presents the study of the gravitational six-body problem which describes the motion of a group of six stars interacting with each other gravitationally. The bodies undergo collisions due to the attractive nature of interactions, which are singular. The singularity is one of the difficulties in exploring the problem. It is necessary to use regularization techniques to avoid singularity when there is a possibility of collisions between the stars. This study involves finding solutions to the six-body problem, especially periodic ones. One important solution to this problem is the Schubart orbit. Schubart orbits were previously discovered for the three-body [52] and four-body [63] problems. We find the Schubart orbit in a setup of six bodies, where the six bodies are positioned at the vertices of two aligned equilateral triangles. The existence proof of the Schubart orbit is also presented. Later, we extend our study of the six-body system to the case, where the bodies are placed on the vertices of two non-aligned equilateral triangles. In this case, we generate families of periodic orbits when the bodies have equal masses. One of the featured results of this problem is the Schubart family of periodic orbits. This family starts from the Schubart orbit and when we move along the family the orbits become more complex and unstable. Another equal-mass family of periodic orbits is generated from the hexagonal orbit. These families of periodic orbits are parametrized by angular momentum. In general, the masses are chosen equal to unity, and the total energy of the system is fixed as E = −1. We perform a non-linear stability check of the discovered solutions under the symmetric perturbations. An algorithm for the search of periodic orbits is presented.
Applied Mathematics