Periodic solutions of radially symmetric perturbations of Newtonian systems

Abstract

Abstract. The classical Newton equation for the motion of a body in a gravitational central field is here modified in order to include periodic central forces. We prove that infinitely many periodic solutions still exist in this case. These solutions have periods which are large integer multiples of the period of the forcing and rotate exactly once around the origin in their period time.

Periodic motions in a gravitational central field with a rotating external force

“…Similar problems have already been considered by the authors of this paper. Radially symmetric systems are studied in [15,16,17] by using topological methods. In this case, the analysis is simplified by the fact that the radial coordinate is ruled by a scalar equation.…”

Periodic orbits of a singular superlinear planar system

“…have been provided in [8][9][10][11][12] in a systematic way. In particular, the case of an attractive singularity was studied in [8,10] and the case of a repulsive singularity was studied in [9].…”

“…In particular, the case of an attractive singularity was studied in [8,10] and the case of a repulsive singularity was studied in [9]. In [11], a singular planar system admitting a superlinear growth was studied.…”

Periodic solutions of singular radially symmetric systems with superlinear growth

“…Recently, exploiting the radial symmetry of system (1), the existence of infinitely many periodic solutions was proved in [18][19][20][21], both in the attractive and in the repulsive case. The main idea there was to split the system into its radial and angular component and to consider the (scalar) angular momentum as a parameter.…”

“…In this paper, we will combine this method with a perturbation argument introduced in [20], to find infinitely many periodic solutions of (1) with a prescribed number of oscillations. In order to do this, we will need to assume that the function e(t, r ) is even in t, so to reduce the study of the radial equation to a Neumann problem, for which the topological degree will be proved to be different from zero.…”