Trouble with the proof for Nielsen's form of Lagrange's equation.

Question

I just cannot understand how the step transition happened. Please please help.

Answer 1

Differentiate each of the three terms on the right with respect to $\dot{q_j}$:

Since $T$ does not depend on $\ddot{q_i}$, $$ \frac{\partial}{\partial\dot{q_j}}\left( \frac{\partial T}{\partial\dot{q_i}} \ddot{q_i}\right) = \frac{\partial^2\,\,T}{\partial q_i\partial q_j} $$

However, since $T$ does depend on both $q_j$ and $\dot{q_j}$, the second term will lead to two contributions: $$ \frac{\partial}{\partial\dot{q_j}}\left( \frac{\partial T}{\partial q_i}\dot{q_i} \right) = \frac{\partial^2 T}{\partial \dot{q_j}\partial q_i}\dot{q_i}+ \frac{\partial T}{\partial q_i} \frac{\partial\dot{q_i}}{\partial\dot{q_j}} = \frac{\partial^2 T}{\partial \dot{q_j}\partial q_i}\dot{q_i}+ \frac{\partial T}{\partial q_i} \delta_{ij} = \frac{\partial^2 T}{\partial \dot{q_j}\partial q_i}\dot{q_i}+ \frac{\partial T}{\partial q_j} $$ And the third term is obvious: $$ \frac{\partial}{\partial\dot{q_j}}\left( \frac{\partial T}{\partial t}\right) = \frac{\partial^2T}{\partial\dot{q_j}\partial t} $$