classical mechanics - 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?

Answer

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.