What is $\frac{d}{dx}\left(\frac{dx}{dt}\right)$? This question was inspired by the Lagrange equation, $\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0$. What happens if the partial derivatives are replaced by total derivatives, leading to a situation where a function's derivative with respect to one variable is differentiated by the original function?

$\frac{\partial L}{\partial \dot{q}}$ may be understood as the generalized momentum of the system (e.g. $\frac{d}{dv}\left(\frac{1}{2}mv^2\right) = mv$, the derivative of kinetic energy with respect to velocity is momentum).  Then $\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}$ is the time derivative of the generalized momentum.  Likewise the $\frac{\partial L}{\partial q}$ term behaves like a generalized force. This begins to look like Newton's second law, but for generalized coordinates.  
