Some theoretical and numerical aspects of the N-body problem
Abstract: 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.
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