This thesis develops methods to identify periodic solutions to the n-body problem by representing gravitational orbits with Fourier series. To find periodic orbits, a minimization function was developed that compares the second derivative of the Fourier series with Newtonian gravitation acceleration and modifies the Fourier coefficients until the orbits match. Software was developed to minimize the function and identify the orbits using gradient descent and quadratic curves. A Newtonian gravitational simulator was developed to read the initial orbit data and numerically simulate the orbits with accurate motion integration, allowing for comparison to the Fourier series orbits and investigation of their stability. The orbits found with the programs correlate with orbits from literature, and a number remain stable when simulated. / February 2016