I was reading L&L and came across
I dont understand how $\frac {\partial L}{\partial {\bf v}} = 0$ implies ${\bf v}$ vector is constant. I tried writing
$$\frac {\partial L}{\partial v} = \frac {\partial L}{\partial vx} i + \frac {\partial L}{\partial vy} j + \frac {\partial L}{\partial vz} k = 0,$$
$$ \frac {\partial L}{\partial vx} i = \frac {\partial L}{\partial vy} j = \frac {\partial L}{\partial vz} k = 0.$$
Therefore $$L(v^2) = c_1 + f(vx,vy) = c_2 + g(vz,vy) = c_3 + h(vz,vx) = c_4$$ Hence $ L(v^2) = const$
However this doesn't imply that the ${\bf v}$ vector is a constant. Can anyone explain what is my mistake in my rough attempt.
