The non-linear Coupled Pendulum

The equations for the coupled pendulum as presented in Hairer et al are: 

2^nd order ODE’s can be numerically integrated by converting into 1^st order equivalents. This can be performed by a straight-forward conversion, or using the State-Space approach. Once represented as 1^st order ODE’s, a numerical technique such as Runge-Kutta or Rosenbrock can be easily implemented to achieve the required output. Shown below is the solution to the coupled pendulum, integrated using the Generalized Rosenbrock method with the VELDS coefficients.