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I've been looking for a term online but couldnt find it: suppose we have a subset $X$ in phase space, such that for all $q\in X$, the path starting at $q$ never exits $X$ either forward or backward in time.

Is there a name for such an $X$?

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A lagrangian submanifold iswhat you are thinking of.

https://ncatlab.org/nlab/show/Lagrangian+subspace

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