Master of Science (MSc)
Mathematics for Science and Finance
Faculty of Arts
The n-body problem models a system of n-point masses that attract each other via some binary interaction. The (n + 1)-body problem assumes that one of the masses is located at the origin of the coordinate system. For example, an (n+1)-body problem is an ideal model for Saturn, seen as the central mass, and one of its outer rings. A relative equilibrium (RE) is a special solution of the (n+1)-body problem where the non-central bodies rotate rigidly about the centre of mass. In rotating coordinates, these solutions become equilibria.
In this thesis we study dynamical aspects of planar (4 + 1)-body systems with the outer four of equal point mass interacting via a modified gravitational “J_2” attraction potential of the form:
V (r) = −((m_i*m_j)/r)*(1 + ϵ/r^2), 0 < ϵ << 1
where r denotes the distance between masses m_i and m_j.
Using the rotational symmetry as well as the symmetry induced by the mass equality, we are able to describe qualitatively the phase-space of square-shaped dynamics. We further analyze the existence of square-shaped RE and establish conditions necessary for their stability. We find that the RE appear as a saddle-node bifurcation. For a unit central mass and equal outer bodies of mass m, we find that these RE are unstable.
Gauthier, Ryan, "Dynamical Aspects in (4+1)-Body Problems" (2023). Theses and Dissertations (Comprehensive). 2559.