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Differential equations describing the dynamics of celestial bodies (rotation, nutation, precession and other effects) can be solved numerically, which makes it possible to study their general non-linear behavior.

My question is this: suppose we want to study the change in time of a planet's orbit with a specific time frame 100-1000 years and at the same study the change in time of its nutation with a period of 0.1-0.5 years. What is the general approach to computing and representing such different time-scale processes?

Remark: In the article The Influence of the Planets, Sun and Moon on the Evolution of the Earth's Axis provides a picture Figure 3. Action of the Sun on the Earth’s rotation. Plot show precession rates and nutation amplitudes on a large time-frame. It can be seen that if these processes differ by orders at time-scale, plot become extremely unreadable. Should we then use a hierarchy of dynamic effects over time for greater clarity?

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    $\begingroup$ This is a very broad question. There are multiple books on this subject. There are undergraduate and graduate level classes devoted to this subject. There are multiple scientific journals that are at least in part devoted to this subject. $\endgroup$
    – David Hammen
    Commented Feb 1, 2023 at 13:30
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    $\begingroup$ What do you mean by "long term"? A thousand years? 30 thousand? A million? 30 million? A billion? Beyond this is well into chaotic behavior, so it gets dicey. $\endgroup$
    – David Hammen
    Commented Feb 1, 2023 at 13:46
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    $\begingroup$ Those are very different problems, demanding very different techniques. $\endgroup$
    – David Hammen
    Commented Feb 1, 2023 at 13:49
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    $\begingroup$ I suggest narrowing the scope of this question to a specific time frame and either to orbital (e.g. precession) or rotational changes (e.g. nutation). As is, this question is too big in scope and is subject to closure. $\endgroup$
    – Connor Garcia ♦
    Commented Feb 1, 2023 at 16:37
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    $\begingroup$ @ConnorGarcia I disagree with you and DH, there can certainly be an answer that explains the differences in dynamical calculation techniques between short timescale and long timescale. Short would be direct numerical, long would be more like perturbation/evolution of the mean orbital elements. Instead of handwringing one should welcome this question; it can have a clear, concise and well-sourced, fact-based answer. $\endgroup$
    – uhoh
    Commented Feb 3, 2023 at 21:36

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