Analytical Solutions for a Nonlinear Coupled Pendulum

The inverse scattering transform is applied to solve the nonlinear equations that govern the motion of two pendulums coupled by an elastic spring. The thetafunction representation of the solutions is describable as a linear superposition of Jacobean elliptic functions (cnoidal vibrations) and additional terms, which include nonlinear interactions among the vibrations. Comparisons between the cnoidal solutions and the solutions obtained by the fourth-order Runge-Kutta scheme are performed. Finally, an interesting phenomenon is put into evidence with consequences for dynamic of the coupled pendula