Sorry for my English (it's not my mother language).
I'm self-studying Mechanics and I'm have a little problem :
We can see that in Landau's book or in Wikipedia that when we inject the Lagrangian in Euler-Lagrange equation the term $\frac{\partial v²}{\partial q}$ vanishes. So we get $$\frac{\partial L}{\partial q}= – \frac{\partial U}{\partial q}$$
here more details :
We want to proof that Euler-lagrange equation imply Newton's second law :
Euler-Lagrange equation state that
$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q}$$
And we set that, for conservative potential fields $ L= T-V(q)= \frac{1}{2}mv² – V(q) $
But if we inject L in Euler-Lagrange equation we will get
$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = m\frac{dv}{dt} – \frac{d}{dt}\frac{\partial V}{\partial \dot{q}}$$
And
$$\frac{\partial L}{\partial q} = \frac{1}{2}m\frac{\partial v²}{\partial q} + F$$
In Landau's book the terms $\frac{\partial V}{\partial \dot{q}}$ and $ \frac{1}{2}m\frac{\partial v²}{\partial q}$ vanishes without any explanation! Then I'm asking:
Why these terms vanish?
Best Answer
This is quite simple. To better visualize Newton's law let's not use generalized coordinates, instead, let's use Cartesian ones. Euler-Lagrange equations hence stands: $$ \frac{d}{dt}\frac{\partial L}{\partial\dot x} - \frac{\partial L}{\partial x} = 0 $$
Since $L = T - V$, it is clear to know that: $$ \frac{\partial L}{\partial x} = \frac{\partial T}{\partial x} - \frac{\partial V}{\partial x} = -\frac{\partial V}{\partial x} \quad\Longrightarrow\quad \frac{\partial L}{\partial x} = -\frac{\partial V}{\partial x} = F $$
Now, on the other term, the potential doesn't depends on the velocity. Hence: $$ \frac{\partial L}{\partial \dot x} = \frac{\partial T}{\partial \dot x} - \frac{\partial V}{\partial \dot x} = \frac{\partial T}{\partial \dot x} \quad\Longrightarrow\quad \frac{\partial L}{\partial \dot x} = \frac{\partial T}{\partial \dot x} = p $$
This can be easily seen: $$ T = \frac{1}{2}m\dot x^2 \Longrightarrow \frac{\partial T}{\partial \dot x} = m\dot x = p $$
Taking those results and plugging into Euler-Lagrange equations, we have: $$ \frac{dp}{dt} - F = 0 $$
Which are, Newton's law.