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I was doing my tutorial on Lagrange-Euler formulation for robotic systems when i came across a slight problem. Referring to the picture in the link, I would like to know if my answer (equation 1) (more specifically my method of differentiating) is correct, given that $\theta_1$, $\theta_2$, $\dot{\theta}_1$, and $\dot{\theta}_2$ are all functions of $t$.

Link to picture:

Thanks a lot!

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A very short answer: the computations in the picture are correct. All you need is to compute the derivatives of the Lagrangian w.r.t. to $\theta_2$ and $t$ to arrive at the Euler Lagrange equations of the 2-body system.

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