Dynamics and integrability of the swinging Atwood machine generalisations

This paper studies the dynamics and integrability of two generalisations of a 3D Swinging Atwood's Machine with additional Coulomb's interactions and Hooke's law of elasticity. The complexity of these systems is presented with the help of Poincare cross sections, phase-parametric diagrams and Lyapunov exponents spectrums. Amazingly, such systems possess both chaotic and integrable dynamics. For the integrable cases we find additional first integrals and we construct general solutions written in terms of elliptic functions. Moreover, we present bifurcation diagrams for the integrable cases and we find resonance curves, which give families of periodic orbits of the systems. In the absence of the gravity, both models are super-integrable.