Double Pendulum Nightmare

Hi all >> see attached images of pendulum and 2nd order model

Currently working on a problem, creating a model for a double pendulum system, we have been given the initial non-linear system of 2x 2nd order equations which have to simplified to 4x 1st order non linear equations; I know the basic theory and know that I can simplify the equations by compacting the variables and going from there

ie. representing them as,

aO1'' + bO2'' + c = 0
dO1'' + eO2'' + f =0

when O1'' = 2nd derivative of phi/ angular acceleration

just a bit lost in how to reproduce the final equations? (Wondering)

The equations are crazy but the theory should be basically the same, would appreciate any ideas......

I'd write:

$\displaystyle y=\dot{\phi_1}$

$\displaystyle z=\dot{\phi_2}$

$\displaystyle k_1y'+k_2\cos(\phi_1-\phi_2)z'+k_2\sin(\phi_1-\phi_2)z^2+k_3\frac{g}{l}\sin(\phi_1)=0$

$\displaystyle k_2\cos(\phi_1-\phi_2)y'+k_5 z'-k_2\sin(\phi_1-\phi_2)y^2+k_2 g\sin(\phi_2)=0$

may want to double check that. Then assign all the $\displaystyle k's$ and then solve it numerically for $\displaystyle y,z,\phi_1,\phi_2$: it's cake using Mathematica's $\displaystyle \text{NDSolve}$ (I think).