I am a student who is very new to mechanics. I have to calculate the Euler-Lagrangian equation for a double pendulum, which is okay. But the angle of the the second pendulum is measured with respect to the first pendulum, and not the vertical. In this case, how do I proceed? Since we need to calculate it with respect to the two angles as coordinates.
The Euler-Larange equations don't care what coordinates you use. That is the beauty of them. You just need to be able to compute the kinetic and potential energies of the system based on the coordinates you choose. Here, the potential of the mass connected to the support is $mgL_1(1-cos(\theta_1))$, where $L_1$ is the length of the rod supporting mass $1$. You need to compute the potential energy of the second mass in terms of $\theta_1$ and $\theta_2$ and the kinetic energies of both masses using the time derivatives of your variables. Once you have those, you plug them into the Euler-Lagrange equations and get differential equations in your variables. The good thing about this approach is that it works for any variables that specify the state of the system and allow you to compute the kinetic and potential energies. This lets you take advantage of symmetries n the system. Often the equations separate more easily if you choose the right variables.