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Once in a book ( I think title was Mathematical Physics, no idea who the author was) saw the proof of instability of planetary orbits in 4 dimensions, cant remember the details of derivations only the final conclusion of the result.

Been trying to do something similar myself, but still with no success, how ever pondered if there is any reading material that shows for dimensions higher than 3 the orbits become unstable.

I remember there were problems that were easier to solve for general cases than the low specific ones. So maybe there is a simple trick that would take care of that problem.

My question is are there any higher dimensions were the orbits remain stable? Is this a mathematical curiosity or was there any body knowledge in physics that considered such cases but ultimately came to conclusion that there is nothing worth pursuing in these lines of thinking?

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