The 3-dimensional cored and logarithm potentials: Periodic orbits

© 2014 AIP Publishing LLC. We study analytically families of periodic orbits for the cored and logarithmic Hamiltonians with 3 degrees of freedom, which are relevant in the analysis of the galactic dynamics. First, after introducing a scale transformation in the coordinates and momenta with a parameter ε, we show that both systems give essentially the same set of equations of motion up to first order in ε. Then the conditions for finding families of periodic orbits, using the averaging theory up to first order in ε, apply equally to both systems in every energy level H = h > 0 showing the existence of at least 3 periodic orbits, for ε small enough, and also provides an analytic approximation for the initial conditions of these periodic orbits. We prove that at every positive energy level the cored and logarithmic Hamiltonians with 3 degrees of freedom have at least three periodic solutions. The technique used for proving such a result can be applied to other Hamiltonian systems.