### Abstract:

This thesis involves the study of a repulsive-attractive N-body problem, which
is a subclass of a quasihomogeneous N-body problem [5]. The quasihomogeneous
N-body problem is the study of N point masses moving in R3N, where the negative
of the potential energy is of the form,
X 1≤i<j≤N
bmimjr−β
ij + X 1≤i<j≤N
amimjr−α
ij .
In the above equation, rij is the distance between the point mass mi and the point
mass mj , and a, b, α > β > 0 are constants. The repulsive-attractive N-body
problem is the case where a < 0 and b > 0.
We start the ground work for the study of the repulsive-attractive N-body
problem by defining the first integrals, collisions and pseudo-collisions and the collision
set. By examining the potentials where a < 0 and b > 0, we see that the
dominant force is repulsive. This means that the closer two point masses get the
greater the force acting to separate them becomes. This property leads to the main
result of the first chapter: there can be no collisions or pseudo-collisions for any
repulsive-attractive system.
In the next chapter we study central configurations of the system. Quasihomogeneous
potentials will have different central configurations than homogeneous
potentials [6], thus requiring the classification of two new subsets of central configurations.
Loosely speaking, the set of central configurations that are not central
configurations for any homogeneous potential are called extraneous. The set of
configurations that are central configurations for both homogeneous potentials that
make up the quasihomogeneous potential, are called simultaneous configurations.
We also notice that every simultaneous central configuration will be non-extraneous,
therefore the two subsets are disjoint.
Next we show the existence of oscillating homothetic periodic orbits associated
with non-extraneous configurations. Finally in this chapter, we investigate the polygon
solutions for repulsive-attractive N-body problems [11]. In particular we show
that the masses need no longer to be equal, for repulsive-attractive potentials. It
will be shown that there exists a square configuration with m1 = m2 6= m3 = m4,
that leads to a relative equilibrium. Therefore, for N = 4 the set of extraneous
configurations is non-empty.
The last chapter deals with the complete analysis of the generalized Lennard-
Jones 2-body problem. The generalized Lennard-Jones problem is the subcase of
the repulsive-attractive N-body problem, where a = −1, b = 2, and α = 2β. We
proceed as in [13] by using diffeomorphic transforms to get an associated system
thereby generating a picture of the global flow of the system. This gives us the
complete flow for the generalized Lennard-Jones 2-body problem.