Analysis of Swing Oscillatory Motion

Historically, nonlinear Hamiltonian systems were studied in many different directions: i) finding of integrable cases and their analytical solutions; ii) investigating the algebraic nature of the integrability; iii) topological analysis of integrable systems and looking for symmetries and so on. The aim of this paper is to find new integrable case(s) of a Hamiltonian system with two degrees of freedom describing the rider and the swing pumped (from the seated position) as a compound pendulum. In result of our analytical calculations we can conclude that this system has two integrable cases when: both dumbbell lengths and point-masses are equal; the gravitational force is neglected.