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We apply the averaging theory of first order to study analytically families of periodic orbits for the cored and logarithmic Hamiltonians HC=12px2+py2+1+x2+y2q2−1, and HL=12px2+py2+12log1+x2+y2q2, which are relevant in the study of the galactic dynamic. We first show, after introducing a scale transformation in the coordinates and momenta with a parameter ɛ, that both systems give essentially the same set of equations of motion up to first order in ɛ. Then the conditions for finding families of periodic orbits, using the averaging theory up to first order in ɛ, apply equally for both systems in every energy level H = h > 0 with H either HC or HL. We prove the existence of two periodic orbits if q is irrational, for ɛ small enough, and we give an analytic approximation for the initial conditions of these periodic orbits. Finally, the previous periodic orbits provide information about the non-integrability of the cored and the logarithmic Hamiltonian systems.

The cored and logarithm galactic potentials: Periodic orbits and integrability