Lagrangian Equations of Motion, Conservative Forces I'm new to this topic so please bear with me. Here on wikipedia we have the Lagrangian equations of motion:
$$ \frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right) - \frac{\partial T}{\partial q} = F_q \label{a}\tag{1} $$
Where $\ T $ is the kinetic energy of the system. A little farther down on the wikipedia page we see the Euler-Lagrange equation (which is the equation I'm currently familiar with):
$$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = F_q \label{b}\tag{2} $$
Where the Lagrangian $\ L $ is
$$ L = T - V \tag{3} $$
And $\ V $ is potential energy.
What is the difference between \ref{a} and \ref{b}? It seems to have something to do with conservative forces but I'm having trouble connecting the dots here. When would it be appropriate to use one equation instead of the other?
 A: The generalised Lagrange equations are
$$
\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{\partial \dot{q}_j} - \frac{\partial T}{\partial q_j}=Q_j \tag{1}
$$
where $T$ is the kinetic energy of the system and $Q$ is the generalised force. This is the most general EoM, and is equivalent to Newton's $F_j=m\ddot x_j$.
Now, if the generalised force can be written as
$$
Q_j = \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial V}{\partial \dot{q}_j} - \frac{\partial V}{\partial q_j} \tag{2}
$$
then, we can plug this into $(1)$:
$$
\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{\partial \dot{q}_j} - \frac{\partial T}{\partial q_j}=\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial V}{\partial \dot{q}_j} - \frac{\partial V}{\partial q_j} \tag{3}
$$
If we define $L\equiv T-V$,  $(3)$ can be rewritten as
$$
\frac{\partial L}{\partial q_j} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}_j}  = 0 \tag{4}
$$
and we get the Euler-Lagrange equations.
To sum up: the most general expression is $(1)$, which is true for any force $Q$. In the special case where $Q$ can be written as $(2)$, then we get the simplified form $(4)$. The important point is that $(2)$ is always true for the relevant forces you'll study, which means that $(4)$ is the important equation, the one that you must remember.
Note that $(2)$ pretty much looks like the condition for a conservative force
$$
F_i=-\frac{\partial V}{\partial q^i}\tag{5}
$$
and in fact is more general than it: it include potentials that can depend on velocity, such as the interaction of charged particles with the electromagnetic field.
