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Documentation for the resolution of Arnold's equation #2375
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Separate out the code changes from #2375
documentation/Celledoni.tex
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@@ -108,7 +110,27 @@ \section*{Solution of Euler's equation} | |||
value of $T$ .\label{figGT}} | |||
\end{figure} | |||
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In the rest of this section, we derive (corrected) formul{\ae} for the three cases described above. | |||
In the rest of this section, we describe our notation and derive (corrected) formul{\ae} for the three cases described above. |
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We are using XeTeX, and it is not the nineties anymore; just write ӕ.
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Done.
documentation/bibliography.bib
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journaltitle = {Monthly Notices of the Royal Astronomical Society}, | ||
number = {4}, | ||
pages = {3620--3632}, | ||
title = {Orientation and rotational parameters of asteroid 4179 Toutatis: new insights from Chang′e-2's close flyby}, |
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Chang PRIME ′
e?
Replace that with an APOSTROPHE '
, which LaTeX will turn into the more appropriate RIGHT SINGLE QUOTATION MARK ’
(or we could directly use ’
everywhere but this is best done in a separate pull request).
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Done, but note that the PRIME
comes from the article title.
documentation/Celledoni.tex
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\cite{Celledoni2007} uses a dimensionless formulation where $\norm\vm = 1$, and absolute values for $I_{jh}$ and $\gD_j$. | ||
We prefer to use a dimensionful formulation where $\norm\vm = G$, and to avoid absolute values. Thus we define: | ||
\begin{align*} | ||
I_{jh} &\DefineAs I_j - I_h &\gD_j &\DefineAs G^2 - 2 T I_j &B_{jh} &\DefineAs \sqrt{±\frac{I_j \gD_h}{I_{jh}}} \\ |
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Add a rad2 in the definition of 𝛥𝑗.
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Done.
documentation/Celledoni.tex
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B_{31}^2 + B_{13}^2 = \frac{\gD_1 I_3 - \gD_3 I_1}{I_{31}} = G^2 | ||
\] | ||
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Physically, $I_{jh}$ has the dimension of an inertial momentum $\squareBrackets{L^2 M}$. $G$ has the dimension of an angular |
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s/inertial momentum/moment of inertia/.
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Done.
momentum $\squareBrackets{L^2 M T^{-1} A}$. $T$ has the dimension of an energy $\squareBrackets{L^2 M T^{-2}}$. | ||
$\gD_j$ has the same dimension as $G^2$. $B_{jh}$ has the same dimension as | ||
$\sqrt{\gD_h}$, i.e., the same dimension as $G$. $\gl_1$ and $\gl_3$ have the | ||
same dimension as the quotient $\frac{G}{I_j}$, i.e., $\squareBrackets{T^{-1} A}$ which is appropriate for their usage. | ||
\subsection*{Case (i)} | ||
Case (i) of the solution of Euler's equation in section 2.2 of \cite{Celledoni2007} is: | ||
\[ |
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Lines 157 and 158 below,
https://github.com/mockingbirdnest/Principia/pull/2375/files#diff-984cb9428da24a818b1049efb3001df7R157-R158,
𝛥3 should be -𝛥3.
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Done.
to map $\vm$ onto $\VectorSymbol{e_1}$ or $\VectorSymbol{e_3}$. In the rest of this section we detail the calculations used to compute $\mathscr S$, | ||
$\mathscr P_t$ and $\gy\of{t}$. | ||
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With the introduction of $\mathscr S$, we are effectively introducing a new base $\mathscr B^p$ for the ``preferred'' principal axes of the body, and |
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Explicitly mention that, with 𝒮, 𝜎 = 1 (and likewise 𝜎′, 𝜎″) from now on; the only allusion to that above is specific to m3 and case (ii).
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Done.
documentation/Celledoni.tex
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\] | ||
and, for the angle $\gy\of{t}$ of the rotation $\mathscr Y_t$ around $\VectorSymbol{e_1}$: | ||
\[ | ||
\TimeDerivative{\gy}\of{t} = \frac{2 T + G m_1 / I_1}{G + m_1} + 4 G \frac{p_1 \TimeDerivative{p_0} - p_0 \TimeDerivative{p_1}}{G + m_1} |
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rad2
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After T?
documentation/Celledoni.tex
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\] | ||
and the angle to: | ||
\[ | ||
\TimeDerivative{\gy}\of{t} = \frac{2 T + G m_1 / I_1}{G + m_1} = \frac{G^2 - \gD_1 + G m_1}{I_1\pa{G + m_1}} = \frac{G}{I_1} - \frac{\gD_1/I_1}{G + m_1} = |
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rad2
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After T?
\[ | ||
\gy\of{t} = \frac{G}{I_1}t + \frac{G I_{13}}{\gl I_1 I_3} | ||
\EllipticPi\of{\JacobiAmplitude\of{\gl t - \gn}, \frac{I_1 I_{32}}{I_3 I_{12}}, k} - | ||
\InverseTrigonometricTangent\of{\sqrt\frac{I_2 I_{31}}{I_3 I_{21}} \JacobiSC\of{\gl t - \gn, k}} |
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s/𝑘/𝑘-1/
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Same.
documentation/Celledoni.tex
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Using equation (\ref{integraldn}) with $a = B_{13}/G$ and simplifying the various coefficients we obtain: | ||
\[ | ||
\gy\of{t} = \frac{G}{I_1}t + \frac{G I_{13}}{\gl I_1 I_3} | ||
\EllipticPi\of{\JacobiAmplitude\of{\gl t - \gn}, \frac{I_1 I_{32}}{I_3 I_{12}}, k} - |
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s/𝑘/𝑘-1/
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I don't think so, I think you meant around line 560.
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Also, I noticed that the second parameter of am(u, k) was missing in a few places.
retest this please |
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