Prove a derivative

Question

Prove that the derivative of

$$E = \frac12m\mathbf{v}^2 - \frac{GMm}{\|\mathbf{r}\|}$$ is

$$\frac{dE}{dt} = (m\mathbf{a})·\mathbf{v} + \mathbf{v}·\left( \frac{GMm}{\|\mathbf{r}\|^3}\right)\mathbf{r}$$

Anyway that I derive it, I cannot figure out how to end up with dot product of $\mathbf{v}$.

$E$ the total energy of a planet,

$m$ is the mass of the planet,

$\mathbf{v}$ is its speed,

$G$ is the universal gravitational constant,

$M$ is the mass of the sun,

$\mathbf{r}$ is the position vector from the sun to the planet

Answer 1

HINT

We have

• $\frac{d}{dt}\frac12v^2=\frac{d}{dt}\frac12(v \cdot v)=\frac122v\cdot\frac{dv}{dt}$
• $\frac{d}{dt}\frac1{\|r\|}=\frac{d}{dt}\frac1{\sqrt{r\cdot r}}=-\frac12\left(2\frac1{r\cdot r\sqrt{r\cdot r}}r\cdot v\right)=-\frac1{{\|r\|^3}}r\cdot v$