This project aims to study the Hamiltonian dynamics of test particles in Newtonian and relativistic discoidal systems, with emphasis mainly in the dynamics of stars in spiral galaxies (especially in the Milky Way). Our goal is to advance in the description of the Hamiltonian dynamics in these systems, focusing on two different aspects: (i) the regular motion around equilibrium points and stable equatorial periodic orbits, through a third integral of motion determined by physical observables (for off-equatorial orbits), according to the recently obtained results regarding the axially symmetric case, and (ii) the chaotic motion near resonances of the spiral pattern and the relation with the recent discoveries about the Galaxy structure near these radii. The second topic will be analyzed mainly by the "Spectral Analysis Method", developed by the Dynamical Astronomy group of IAG--USP, and successfully applied to planetary systems and asteroid motion. The resemblance between the two formalisms for the motion in planetary systems and in galaxies allows us to apply it also to galactic dynamics problems, since both correspond to the dynamics of a particle subject to an external gravitational potential.As a second part of the project, we intend to apply the aforementioned method to the dynamics of test particles in general relativity, with focus on systems containing bidimensional or three-dimensional disks, using the Hamiltonian formalism for the geodesic flow. The study of chaos in relativistic disk models is recent. Our main goal is to analyze how chaos appears in disk systems with known Newtonian analogue, as well as to verify if the relativistic extensions of Newtonian systems with integrable dynamics present chaos, and in this case to study the reason for this behavior. A last problem is the dynamical analysis of orbits in post-Newtonian disk systems (1PN), following the Hamiltonian formulation for test particles recently developed by the candidate, comparing with the Newtonian case. We expect that the rotation in stationary disks introduces qualitative effects in the phase-space structure already in the 1PN approximation, without the necessity of an exact relativistic approach.
News published in Agência FAPESP Newsletter about the scholarship: