Hello!(adsbygoogle = window.adsbygoogle || []).push({});

Let's say our gravitational potential is (as usual for 2 body),

$$a = -\frac{\mu}{r^3} \mathbf{r}$$.

Then the gradient of this is G,

$$\frac{\partial G}{\partial \mathbf{r}} = G = \frac{\mu}{r^3} [3 \hat{\mathbf{r}} \hat{\mathbf{r}}^\top - I] $$

Now if we take two time derivatives of G, we get

$$ \ddot{G} = \frac{3\mu}{r^5} [\hat{\mathbf{r}}^\top \mathbf{v})^2 (7 \hat{\mathbf{r}} \hat{\mathbf{r}}^\top - I) - 10(\hat{\mathbf{r}}^\top \mathbf{v})(\mathbf{v} \hat{\mathbf{r}}^\top + \hat{\mathbf{r}} \mathbf{v}^\top) + 2 \mathbf{v} \mathbf{v}^\top - (\mathbf{v}^\top \mathbf{v})(5 \hat{\mathbf{r}} \hat{\mathbf{r}}^\top - I) + (\hat{\mathbf{r}}^\top \hat{\mathbf{r}}) G]$$

Also for completeness, $$\mathbf{r}$$ is the position vector. $$\hat{\mathbf{r}}$$ is the unit position vector. $$r$$ is the norm of the position vector. $$ \mathbf{v}$$ is the velocity vector.

Now this is where it gets tricky, I need to take 3 more time derivatives of $$ \ddot{G}$$ so that I have up to the 5th derivative. The problem is, it's getting so long and tedious that I keep making mistakes. Is there a quick way of taking these derivatives? I tried mathematica but it just gets really messy because it does it in components rather than a vector. Any help is appreciated!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Multiple time derivatives of gravitational potential

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads for Multiple derivatives gravitational |
---|

I Astrodynamics Question: Derivation of Sp. Orbital Energy? |

A Second derivative of a complex matrix |

I How to find the matrix of the derivative endomorphism? |

B Associativity of Matrix multiplication |

**Physics Forums | Science Articles, Homework Help, Discussion**