This paper presents a Hamiltonian approach to modelling relative coorbital motion based on derivation of canonical coordinates for the relative coorbital dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations while allowing us to obtain closed-form solutions to the relative coorbital motion in Hill's restricted three-body problem. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton-Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic orbital elements. The influence of the gravitational interaction between the coorbiting satellites is incorporated into the analysis by a variation of parameters procedure.