#### Document Type

Thesis

#### Degree Name

Master of Science (MSc)

#### Department

Mathematics

#### Program Name/Specialization

Mathematics for Science and Finance

#### Faculty/School

Faculty of Arts

#### First Advisor

Cristina Stoica

#### Advisor Role

Supervisor

#### Abstract

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.

#### Recommended Citation

Gauthier, Ryan, "Dynamical Aspects in (4+1)-Body Problems" (2023). *Theses and Dissertations (Comprehensive)*. 2559.

https://scholars.wlu.ca/etd/2559

#### Convocation Year

2023

#### Convocation Season

Spring

#### Included in

Astrophysics and Astronomy Commons, Dynamical Systems Commons, Dynamic Systems Commons, Numerical Analysis and Computation Commons, Ordinary Differential Equations and Applied Dynamics Commons