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?

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.

• Thank you for the detailed answer! I'll accept it when the timer lets me (new account restrictions). In my book Game Physics by Eberly he derives motion for a pendulum with equation (1). It's on pages 101 and 102 (publicly viewable in my Amazon link). Any idea why he used (1) instead of (4), if (4) is the simplified version? Surely forces are conservative here. Commented May 14, 2016 at 21:05
• youre welcome, Im glad I could help :-) I guess Eberly is using (2) because he hasn't introduced (4) yet (the first time we see Euler-Lagrange equations is on page 129). At that point, on pages 101-102 he has only introduced (2), and so he does the example exercise with that equations to show how they work. Maybe after page 129, when he has already explained the meaning of $L$ he'll redo the example, this time with (4) instead of (2). But yes: in that example the force is conservative, so (4) is the best way to approach the problem. Commented May 14, 2016 at 21:14
• For anyone else that comes by, I found an example of pendulum motion with and without gravity noticed as a conservative force. Commented May 14, 2016 at 21:25
• It might be relevant to mention that Eq. (2) is valid for fundamnetal interactions, but not, in general, for frictional forces, which do play a rôle in life. Commented Aug 24, 2018 at 10:48