Tenemos $$J[\phi_1,\phi_2] = \int_{t_0}^{t_1} \left[ab^2 \dot{\phi}_1^2 + \frac{1}{2}ab^2 \dot{\phi}_2^2 + ab^2\phi_1\phi_2\cos(\phi_1 - \phi_2) \right] dt$$ donde $\phi_1=\phi_1(t), \phi_2=\phi_2(t)$ , $a,b$ constante.
Las ecuaciones de Euler-Lagrange para varias funciones de una sola variable son $\displaystyle 0= \frac{\partial f}{\partial \phi_i} - \frac{\partial}{\partial t}\frac{\partial f}{\partial\dot{\phi}_i},$ donde $$f(t,\phi_1, \phi_2, \dot{\phi}_1, \dot{\phi}_2) =ab^2 \dot{\phi}_1^2 + \frac{1}{2}ab^2 \dot{\phi}_2^2 + ab^2\phi_1\phi_2\cos(\phi_1 - \phi_2).$$
Tengo \begin{align} 0 &= \frac{\partial f}{\partial \phi_1} - \frac{\partial}{\partial t}\frac{\partial f}{\partial \dot{\phi}_1}\\ &= ab^2\phi_2\cos(\phi_1 - \phi_2) - ab^2\phi_1\phi_2\sin(\phi_1 - \phi_2) - \frac{\partial}{\partial t}\left( 2ab^2 \dot{\phi}_1 \right)\\ &= ab^2\phi_2\cos(\phi_1 - \phi_2) - ab^2 \phi_1\phi_2\sin(\phi_1 - \phi_2) - 2ab^2\ddot{\phi}_1\\ 0 &= \frac{\partial f}{\partial \phi_2} - \frac{\partial}{\partial t}\frac{\partial f}{\partial\dot{\phi}_2}\\ &= ab^2\phi_1\cos(\phi_1-\phi_2)+ab^2\phi_1\phi_2\sin(\phi_1 - \phi_2)-\frac{\partial}{\partial t}(ab^2 \dot{\phi}_2)\\ &= ab^2\phi_1\cos(\phi_1 - \phi_2)+ab^2\phi_1\phi_2\sin(\phi_1 - \phi_2)-ab^2 \ddot{\phi}_2 \end{align} como el sistema de 2 odas de segundo orden: \begin{align*} 0&=\phi_2\cos(\phi_1 - \phi_2) - \phi_1\phi_2\sin(\phi_1 - \phi_2) - 2\ddot{\phi}_1 \\ 0&=\phi_1\cos(\phi_1 - \phi_2)+\phi_1\phi_2\sin(\phi_1 - \phi_2)- \ddot{\phi}_2. \end{align*}
¿Voy por buen camino? ¿Hay dos ecuaciones EL, una para cada $\phi_j?$
