| dc.description.abstract | In this work, we will be investigating a specific Hamiltonian system, namely, the Kepler problem with a correction term $\frac{\delta}{r^{3}}$ added to the potential energy. Our objective is to show that the system is stable in the sense of the KAM theorem. In the first sections, we introduce essential concepts and tools that will be used in the process of understanding and showing how the KAM theorem works with our system. These concepts and tools are: Hamiltonian formalism, canonical transformations, the Hamilton-Jacobi equation and Action-Angle variables. In the last section, we state the KAM theorem and, based on the results we obtain from previous sections, we can conclude that the system is in fact stable in the sense of the KAM theorem. An informal statement of the KAM theorem is that if the unperturbed Hamiltonian system $H_{0}$, expressed in the action variable $J$, is non-degenerate, then under sufficiently small perturbation $\epsilon H_{1}$ we have that \begin{equation*}\label{pert} H(J,\Phi)=H_{0}(J)+\epsilon H_{1}(J,\Phi) \end{equation*} for $\epsilon 0$. | |
