Why does d/dv(L) = constant imply velocity is constant?

[zhuang382]

Homework Statement

How do we deduce the conclusion of (3,2)? This is a topic inertial frame.

Relevant Equations

Law of inertia

I understand that d/dv(L) = constant, and L is only dependent on v, but how do we get to the fact that v = constant?

 

[anuttarasammyak]

Say
$$\frac{\partial L}{\partial v}=f(v)$$,

$$\frac{d}{dt}f(v)= \frac{df(v)}{dv}\frac{dv}{dt}=0$$

Usually $\frac{df(v)}{dv} \neq 0$ so

$$\frac{dv}{dt}=0$$

 

[zhuang382]

Thanks!