• Keplerian dynamics
• n-body systems
• Numerical integration
• Post-Newtonian approximation
• Sagittarius A*

Compact stellar clusters are potential sources of stochastic gravitational waves due to repeated close encounters between compact stars. A detailed calculation of the expected signal is hitherto missing.
As a first step towards such calculations one have to first implement the basic, classical, physical environment with all the necessary tools like integration procedures, management modules both for the "saving" and "reading" phase of the solution or data analysis functions. Once this is done and classical physics works fine, one can move to implement the more complex block of modern gravitation.

So, with this initial aim, we start introducing the fundamental notions about the two-body gravitational problem. At first, we remember the classical laws of orbital mechanics and then their modern re-interpretation thanks to the work of Albert Einstein. We also show how, in a certain working regime, General Relativity can be linearized in order to obtain laws that are simple to implement. Moreover, in this part we work on the generalization of the routines, extending the integration to an arbitrary number $n$ of corpses.
We then move to explain the concepts behind the various aspect of numerical integration of differential equations, with all the practical problems related to this operation.

After that, thanks to the theoretical notions introduced, we manage to work out various numerical integrators to solve our system of Hamilton equation and thus find the overall evolution of a generic gravitational system. The different aspects and fragilities of these algorithms are also treated.

Once everything has been set, we test all of these integration techniques through various simulations of the same Sun-Earth system in order to have a direct comparison between our numerical solutions and the analytical one. Thanks to these experiments, we have the opportunity to highlight their strengths and weaknesses in order to declare which one of them behaves better in this kind of scenario.
To make sure that the simulations can be executed by almost every calculator, we organized the "saving" and the "reading" phases of the data manageable through a certain number of settable parameters. This makes the whole code executable by the majority of machines without the risk of running out of RAM.

Once we chose the best integrator at our disposal, the project follows with the testing of the code's performance in various well-known scenarios. This "testing" phase is always necessary when a new code is written from zero, in order to be sure that it is reliable.
We will deal with a purely numerical estimation of the Mercury's perihelion shift effect induced by General Relativity and a reproduction of the whole Solar System's dynamic itself.

In the final part of this work, we try to apply the functionalities of our code to a less known system: the reproduction of the innermost stars' motion around Sagittarius A*, the super-massive black hole at the centre of the Milky Way. In this region there must be a very strong gravitational potential, meaning that we're shifting away from our "weak-field" assumptions, with all the consequences that it brings.