I have arrived at the equations of motion for a double pendulum, with gravity $g$, masses $m_i$, link lengths $l_i$, angles $\theta_i$, and applied torques $\tau_i$. Please see the diagram and derivation at Diego Assencio's blog.
I end up with a vector equation in matrix $2\times 2$ matrix $A$ and $2 \times 1$ vector $b$.
$$A(\theta_1, \theta_2) \begin{bmatrix} \ddot{\theta_1} \\ \ddot{\theta_2} \end{bmatrix} + b(\theta_1, \theta_2, \dot{\theta_1}, \dot{\theta_2}) = \begin{bmatrix} \tau_1 \\ \tau_2 \end{bmatrix} $$ See [1] at the bottom of this post for the expanded form.
Solving, we can see how applying exogenous torqes influences the angular accelerations of the system.
$$ \begin{bmatrix} \ddot{\theta_1} \\ \ddot{\theta_2} \end{bmatrix} = A(\theta_1, \theta_2)^{-1}\begin{bmatrix} \tau_1 \\ \tau_2 \end{bmatrix} - A(\theta_1, \theta_2)^{-1} b(\theta_1, \theta_2, \dot{\theta_1}, \dot{\theta_2}) $$
On account by multiplying through by $A^{-1}$, we see that an exogenous torque $\tau_2$ will contribute to both angular accelerations $\ddot{\theta_1}$ and $\ddot{\theta_2}$. This is what the mathematics tells us, but I can't wrap my head around what's physically happening. Supposing I had an actuator in the second joint and I apply torque, how does the first joint know it has to angular-ly accelerate?
[1] $$\ddot{\theta_1} l_{1}^{2} \left(m_{1} + m_{2}\right) + \ddot{\theta_2} l_{1} l_{2} m_{2} \cos{\left(\theta_{1} - \theta_{2} \right)} + \dot{\theta_2}^{2} l_{1} l_{2} m_{2} \sin{\left(\theta_{1} - \theta_{2} \right)} - \tau_{1} + g l_{1} \left(m_{1} + m_{2}\right) \sin{\left(\theta_{1} \right)} = 0$$ $$\ddot{\theta_1} l_{1} l_{2} m_{2} \cos{\left(\theta_{1} - \theta_{2} \right)} + \ddot{\theta_2} l_{2}^{2} m_{2} - \dot{\theta_1}^{2} l_{1} l_{2} m_{2} \sin{\left(\theta_{1} - \theta_{2} \right)} - \tau_{2} + g l_{2} m_{2} \sin{\left(\theta_{2} \right)} = 0$$
