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## Homework Statement

I was reading about the Hermite integration scheme for N-body simulations, as seen here: http://www.artcompsci.org/kali/vol/two_body_problem_2/ch11.html#rdocsect76

This scheme uses jerk, the time rate of change of acceleration. The problem is that I don't know how to correctly evaluate the derivative of the acceleration equation shown there.

## Homework Equations

[tex] \vec{a} = \frac{GM}{r^2}\vec{\hat{r}} [/tex]

[tex] \vec{j} = GM\left(\frac{\vec{v}}{r^3} - 3 \frac{(\vec{r} \cdot \vec{v})\vec{r}}{r^5}\right) [/tex]

## The Attempt at a Solution

First I rewrote the unit vector in

**a**to get:

[tex]\vec{a}= \frac{GM\vec{r} }{r^3}[/tex]

To differentiate, I applied the quotient rule:

[tex]\frac{d\vec{a}}{dt} = GM\frac{r^3\frac{d\vec{r}}{dt} - \vec{r}\frac{dr^3}{dt}}{r^6} [/tex]

The first term is obviously

**v**/r^3, but the second term is where I'm confused. I can get the right answer by using the chain rule:

[tex]\frac{\frac{dr^3}{dt}}{r^6} = \frac{3r^2\frac{d\vec{r}}{dt}}{r^6} = \frac{3\vec{r}\frac{d\vec{r}}{dt}}{r^5} = \frac{3(\vec{r}\cdot\vec{v})}{r^5}[/tex]

but I'm just arbitrarily changing the scalar r into the vector

**r**. What's the right way to do this step?

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