Some theoretical and numerical aspects of the N-body problem

The N-body problem has been studied for many centuries and is still of interest in contemporary science. A lot of effort has gone into solving this problem but it's unlikely that a general solution will be found with the mathematical tools we have today. We review some of the progress that has been made over the centuries in solving it. We take a look at the first integrals, existence of solutions and where singularities can occur. We solve the two body problem and take a look at the special case of central configurations. We find all the possible three-body central configurations, which are known as Euler's and Lagrange's solutions. When analytic solutions are missing it is natural to use numerical methods. We implement and compare four numerical solvers for differential equations: Euler's method, Heun's method, the classical fourth-order Runge-Kutta scheme and Störmer-Verlet. Comparison of accuracy is made using the known solutions discussed in the previous parts of this thesis.The famous physicist and mathematician Sir Isaac Newton formulated the universal gravitation theory about 300 years ago. This theory has allowed scientists to make calculations in order to predict movement of objects under the influence of gravitational force. In astronomy the movement of planets and asteroids can be formulated as a mathematical equation based on Newton's laws of gravitation. The equation describes how a force changes the speed and direction of the moving particles in space. In order to predict a future location one needs to solve this equation. This is referred to as the N-body problem. Still today the general problem is considered unsolved. In this thesis we will consider solutions to special cases of the problem such as when there are only two bodies. We will also take a look at numerical methods that can be used when analytical solutions don't exist