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I am required to rewrite the equation

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

as a first order differential equation. How do I go about doing this?

I understand how this equation was derived from the original

$$\mathbf{F}_{21}=-\frac{G m_1 m_2}{r^2}\hat{\mathbf{u}}_r$$

but I am struggling to find information on the process of attaining $\dot{\mathbf{r}}$ from the above equations.

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  • $\begingroup$ Who is requiring you? If this is a homework question make sure to consider these points. Which are also relevant for answers if this indeed a homework question. meta.stackoverflow.com/questions/334822/… $\endgroup$ Nov 21, 2022 at 13:52

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Given a scalar function $f(x)$ that is described as an $n^{th}$ order differential equation can be converted to a first order $n$ dimensional vector differential equation. For example, suppose $$\begin{aligned} f'(x) &= g(f(x), x) \\ f''(x) &= h(f(x), f'(x), x) \end{aligned}$$ The solution is to make an augmented function $\vec u(x)$ such that $u[1](x) = f(x)$ and $u[2](x) = f'(x)$. The same can be applied to a vector function $\vec r(t)$. In the case of gravitation, we the result is a six vector, where $u[1](t) = r_x(t)$, $u[2](t) = r_y(t)$, $u[3](t) = r_z(t)$, $u[4](t) = v_x(t)$, $u[5](t) = v_y(t)$, and $u[6](t) = v_z(t)$. The time derivatives of the first three elements are simply the velocity vector, while the time derivatives of the last three elements are given by Newton's law of gravitation.

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R double dot is three second-order equations. The standard trick to convert it into six first-order equations is to introduce velocity, so R dot = V and V dot = what R double dot was. Then, writing a six-element state vector as R conjoined with V, you can write the six equations in matrix form as something like

$$\begin{bmatrix}\dot{r}_x\\ \dot{r}_y\\ \dot{r}_z\\ \dot{v}_x\\ \dot{v}_y\\ \dot{v}_z\end{bmatrix}= \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ & \cdots & & & & \\ & & \cdots & & & \\ & & & \cdots & & \end{bmatrix} \begin{bmatrix}r_x\\ r_y\\ r_z\\ v_x\\ v_y\\ v_z\end{bmatrix} $$

Filling in the blanks may require some rearrangement of variables and application of the chain rule, depending on which coordinates and additional perturbing terms you are using.

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    $\begingroup$ That matrix form is a linearization of the nonlinear 2nd order gravitational equation. It does not answer the question. $\endgroup$ Nov 20, 2022 at 18:58

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