# Equilibrium Distribution of Charges, Capacities, and Affine Mappings

## Abstract

In this dissertation, we will explore the interactions of potential charges within simple geometric domains. The positions of these charges in reaching an extremal position will then be transitioned to a study on the interaction of the charges based on the shape of the domain changing. This will lead into a final study, where a conformal invariant, based on the shape of a particular domain, is examined as the domain is transformed by a complex mapping. The common thread between all of these topics is that a particular energy for a system is studied; the energy of each system, though, is based on the interactions the charges have in their respective domains.

In Chapter 2, we will discuss the historical background of the study of charge placement with regard to minimizing potential energy. Some of the applications of potential energy in other fields will be discussed as well.

In Chapter 3, the planar configurations of charges on simple geometric domains will be discussed. Of interest is the placement of charges that produce an extremal logarithmic potential energy based on the domain containing these charges.

In Chapter 4, we will discuss the break-of-symmetry effect with regard to the position charges on domains as the shape of the domains changes. We will show that n extremal charges will have a convergent limit set of n charges. However, for one particular system the critical point of the function describing the system’s logarithmic potential energy will be where a “jump” occurs in the extremal position of the charges.

In Chapter 5, we will discuss the affine capacity of a system of charges. The affine modulus of a quadrilateral is introduced, and the behavior of the affine modulus of two “essentially different” quadrilaterals under an affine transformation will also be discussed.