We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem.

We study a second-order ordinary differential equation coming from the Kepler problem on ${{\mathbb{S}}^{2}}$. The forcing term under consideration is a piecewise constant with singular nonlinearity that changes sign. We establish necessary and sufficient conditions to the existence and multiplicity of $T$-periodic solutions.

The present paper provides a numerical investigation of the decoherence effect induced on a quantum heavy particle by the scattering with a light one. The time dependent two-particle Schrödinger equation is solved by means of a time-splitting method. The damping undergone by the non-diagonal terms of the heavy particle density matrix is estimated numerically as well as the error in the Joos-Zeh approximation formula.

The classical discrete element approach (DEM) based on Newtonian dynamics can be divided into two major groups, event-driven methods (EDM) and time-driven methods (TDM). Generally speaking, TDM simulations are suited for cases with high volume fractions where there are collisions between multiple objects. EDM simulations are suited for cases with low volume fractions from the viewpoint of CPU time. A method combining EDM and TDM called Hybrid Algorithm of event-driven and time-driven methods (HAET) is presented in this paper. The HAET method employs TDM for the areas with high volume fractions and EDM for the remaining areas with low volume fractions. It can decrease the CPU time for simulating granular flows with strongly non-uniform volume fractions. In addition, a modified EDM algorithm using a constant time as the lower time step limit is presented. Finally, an example is presented to demonstrate the hybrid algorithm.