MHB State Matrix Derivation for Orbital Mechanics

AI Thread Summary
The discussion focuses on deriving the equation \(\ddot{\mathbf{r}} = -\frac{\mu}{r^3}\mathbf{r}\) into state matrix form for orbital mechanics. Participants seek clarification on defining Keplerian motion as the divergence of the potential and the implications of the state transition matrix being symplectic. The conversation includes detailed mathematical expressions for the state transition matrix and its components, specifically the matrices \(\mathbf{\Phi}_{ij}\) and their partial derivatives related to initial conditions. Additionally, there are inquiries about the derivation of these matrices and the sensitivities of the semimajor axis with respect to initial state vectors. The discussion is ongoing, with requests for further explanations on specific derivations and concepts.
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Has anyone seen the derivation of \(\ddot{\mathbf{r}} = -\frac{\mu}{r^3}\mathbf{r}\) into state matrix form?

If so, can they provide a link?
 
Last edited:
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In a book, I have found a derivation I don't fully understand.

For one, it defines \(\ddot{\mathbf{r}} = -\frac{\mu}{r^3}\mathbf{r} = -\nabla V(\mathbf{r})\). I have never seen this definition. Why/How can Keplerian motion de defined as divergence of the potential of r?

Then
\[
\mathbf{G} = -
\begin{bmatrix}
\frac{\partial(\nabla V(\mathbf{r}))}{\partial\mathbf{r}}
\end{bmatrix} = -
\begin{bmatrix}
V_{xx} & V_{xy} & V_{xz}\\
V_{xy} & V_{yy} & V_{yz}\\
V_{xz} & V_{zy} & V_{zz}
\end{bmatrix}
\]
Since \(\mathbf{G} = \mathbf{G}^T\), the state transition matrix of the Keplerian two body problem is guaranteed to be symplectic.
\[
\mathbf{x}(t) =
\begin{bmatrix}
\mathbf{r}(t)\\
\mathbf{v}(t)
\end{bmatrix} =
\begin{bmatrix}
F\cdot\mathbb{I} & G\cdot\mathbb{I}\\
\dot{F}\cdot\mathbb{I} & \dot{G}\cdot\mathbb{I}
\end{bmatrix}\mathbf{x}_0
\]
where \(\mathbb{I}\) is \(3\times 3\) and
\begin{align}
F &= 1 - \frac{a}{r_0}(1 - \cos(\Delta E))\\
G &= \Delta t + \sqrt{\frac{a^3}{\mu}}(\sin(\Delta E) - \Delta E)\\
\dot{F} &= -\frac{\sqrt{a\mu}}{rr_0}\sin(\Delta E)\\
\dot{G} &= 1 + \frac{a}{r}(\cos(\Delta E) - 1)
\end{align}
The state transition matrix for this nonliear systme is defined as (Why?)
\[
\mathbf{\Phi}(t,t_0) =
\begin{bmatrix}
\mathbf{\Phi}_{11} & \mathbf{\Phi}_{12}\\
\mathbf{\Phi}_{21} & \mathbf{\Phi}_{22}
\end{bmatrix} =
\begin{bmatrix}
\frac{\partial\mathbf{x}(t)}{\partial\mathbf{x}_0}
\end{bmatrix}
\]
Then it says subdividing the \(6\times 6\) state transition matrix into four \(3\times 3\) matrices \(\mathbf{\Phi}_{ij}\), and using \(F\) and \(G\) solutions to compute the required partial derivatives, leads to the following results: (Can some one walk we through the first one so I understand how these were derived?)
\begin{align}
\mathbf{\Phi}_{11} &= F\cdot\mathbb{I} + \mathbf{r}_0\frac{\partial F}{\partial\mathbf{r}_0} + \mathbf{v}_0\frac{\partial G}{\partial\mathbf{r}_0}\\
\mathbf{\Phi}_{12} &= G\cdot\mathbb{I} + \mathbf{r}_0\frac{\partial F}{\partial\mathbf{v}_0} + \mathbf{v}_0\frac{\partial G}{\partial\mathbf{v}_0}\\
\mathbf{\Phi}_{21} &= \dot{F}\cdot\mathbb{I} + \mathbf{r}_0\frac{\partial \dot{F}}{\partial\mathbf{r}_0} + \mathbf{v}_0\frac{\partial \dot{G}}{\partial\mathbf{r}_0}\\
\mathbf{\Phi}_{22} &= \dot{G}\cdot\mathbb{I} + \mathbf{r}_0\frac{\partial\dot{F}}{\partial\mathbf{v}_0} + \mathbf{v}_0\frac{\partial\dot{G}}{\partial\mathbf{v}_0}\\
\end{align}
The partial derivatives of the initial orbit radius \(r_0\) and velocity magnitude \(v_0\) are given by (Can someone explain this?)
\begin{align}
\frac{\partial r_0}{\partial\mathbf{r}_0} &= \frac{1}{r_0}\mathbf{r}_0^T\\
\frac{\partial r_0}{\partial\mathbf{v}_0} &= \mathbf{0}^T\\
\frac{\partial v_0}{\partial\mathbf{r}_0} &= \mathbf{0}^T\\
\frac{\partial v_0}{\partial\mathbf{v}_0} &= \frac{1}{v_0}\mathbf{v}_0^T
\end{align}
Using the definition of \(\sigma_0\equiv \frac{1}{\sqrt{\mu}}\mathbf{r}_0^T\mathbf{v}_0\), the partial of \(\sigma_0\) is
\begin{gather}
\frac{\partial\sigma_0}{\partial\mathbf{r}_0} = \frac{1}{\sqrt{\mu}}\mathbf{v}_0^T\\
\frac{\partial\sigma_0}{\partial\mathbf{v}_0} = \frac{1}{\sqrt{\mu}}\mathbf{r}_0^T
\end{gather}
To find the sensitivities of the semimajor axis \(a\) with respect to the initial state vectors, we write the energy equation as
\[
\frac{1}{a} = \frac{2}{r_0} - \frac{v_0^2}{\mu}
\]
Then let's introduce a place holder vector alpha.

To be continued... but if you know how to answer anything already asked, please do.
 
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