Given $$2 \vec{\varepsilon }\cdot\frac{\delta L}{\delta v^2}\frac{d \vec{r}}{dt} = \frac{dg(\vec{r},t)}{dt}$$ with $\vec{\varepsilon }\in \mathbb{R}^3$ is infinitesimal small vector, $\vec{r}=\vec{r}(t) \in \mathbb{R}^3$, $\vec{v} = \frac{d\vec{r}}{dt}$, $L = L(v^2)$ a function of $v^2$, and $g(\vec{r},t)$ an arbitrary total differentiable function of time $t$.
show that $\frac{\delta L}{\delta v^2}$ be constant.
I just apply chain rule: $\frac{dg(\vec{r},t)}{dt} = \frac{dg}{d\vec{r}}\frac{d\vec{r}}{dt}$ but see nothing.