I'm in my final year of my undergraduate studies doing work on modelling the n-body problem numerically and I also have some interest in theoretical guarantees. Now, I've been looking for a theorem that guarantees the following:

$$\text{KAM stable} \implies \text{smooth orbits} \tag{*}$$

To be precise I mean that if we decompose a KAM stable solution to the n-body problem into n curves(not necessarily distinct) then each of these curves must be smooth. I haven't come across such a theorem so far but I'm certain it must be true.

Note 1: I haven't read all the literature on KAM theory but I've read a few introductions on the subject(ex. Jacques Fejoz).

Note 2: Smoothness would rule out collision or non-collision singularities but it doesn't imply KAM stability as there are smooth orbits which aren't KAM stable. Ex: all solutions to the three body problem besides the figure eight solution. So the converse is obviously false.

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