Periodic orbits in the logarithmic potential

Abstract

Context. We investigate periodic orbits in galactic potentials by developing analytical methods. Aims. We evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. Methods. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series, a resummation based on a continued fraction may be performed. This method is analogous to the Prendergast method, which searches for approximate rational solutions. Results. It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct accurate solutions both for normal modes and periodic orbits in general position.

On the integrability of a three-dimensional cored galactic Hamiltonian

“…The motivation for the choice of the potential 1 + x 2 + y 2 + z 2 /q comes from the interest of this potential in galactic dynamics, see for instance [2,3,5,6,7,9,12,13,14,15,16]. The parameter q gives the ellipticity of the potential, which ranges in the interval √ 0.6 ≤ q ≤ 1.…”

On the analytic integrability of the cored galactic Hamiltonian

“…The potential √ 1 + x 2 + y 2 q 2 has an absolute minimum and a reflection symmetry with respect the two axis x and y. The motivation for the choice of these symmetries comes from the interest of this potential in galactic dynamics, see for instance [2,3,4,9,10,11]. The parameter q gives the ellipticity of the potential, which ranges in the interval 0.6 ≤ q ≤ 1.…”

The cored and logarithm galactic potentials: Periodic orbits and integrability

“…al. [12] to find periodic orbits. The structure of the phase space related to the logarithmic potential has been approximated with resonant detuned normal forms constructed with the method based on the Lie transform by Pucacco et.al.…”

“…Then there exist the two periodic orbits given by Theorems 1 and 2. Moreover their associated Jacobians (12) are different from 1, in fact, they go to infinity. Since the Jacobians are the product of the multipliers of these periodic orbits, all the multipliers cannot be equal to 1.…”