Question in Lagrangian formalism

Question

In lagrangian mechanics, where $L=T-U$ and the lagrangian formulation is $ \frac{d}{dt}\big( \frac{\partial L}{\partial \dot{q_i}}\big)-\frac{\partial L}{\partial {q_i}}=F_i$, where $F$ is the non-conservative force.

My question is if I want to find out the above equation for a given problem then the $q_i$ should be written for every term in which the system is expressed. Like if I want to write the equation for a pendulum then the $q_i$ will be the angle displacement.

So for example in a double pendulum there will be two angles $\phi ,\theta $ for the respective rods than the equation in lagrangian formalism will be $$\frac{dL}{dt}(\frac{\partial L}{\partial \dot{\phi}}+\frac{\partial L}{ \partial \dot{\theta}})+\frac{\partial L}{\partial \theta}+\frac{\partial L}{\partial \phi}=0$$ is this correct?

Answer 1

No, you get a separate Euler-Lagrange equation for each individual degree of freedom, i.e. a system of simultaneous equations. So in your example,

\begin{align} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\theta}}\right)-\frac{\partial L}{\partial \theta} &= 0, \,\mathrm{and} \\ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right)-\frac{\partial L}{\partial \phi} &= 0 \end{align}