Integrability in a Nonlinear Model of Swing Oscillatory Motion

Nonlinear dynamical systems can be studied in many different directions: i)~finding integrable cases and their analytical solutions, ii)~investigating the algebraic nature of the integrability, iii)~topological analysis of integrable systems, and so on. The aim of the present paper is to find integrable cases of a dynamical system describing the rider and the swing pumped (from the seated position) as a compound pendulum. As a result of our analytical calculations, we can conclude that this system has two integrable cases when: 1)~the dumbbell lengths and point-masses meet a special condition; 2)~the gravitational force is neglected.