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UPD: I would also add a third possibility, namely that one of the configurations can be unstable. That is, it will be a valid solution of the equations, but any small variation will lead to the system leaving that configuration. Moreover, this means that such a configuration will not be reachable in the first place. You did not do any stability analysis, but if your manual experiment shows that both configurations are possible, then most probably both are stable.

(Not to mention that the required "very high precision", if for example we talk about spatial position, can easily be much less than the size of an atom, which is beyond the limits of applicability of classical mechanics. That it, the result may radically change if you change the initial position by a fraction of atom size; but you can not require a classical mechanics initial state to be specified to that precision.)

(Not to mention that the required "very high precision", if for example we talk about spatial position, can easily be much less than the size of an atom, which is beyond the limits of applicability of classical mechanics. That is, the result may radically change if you change the initial position by a fraction of atom size; but you can not require a classical mechanics initial state to be specified to that precision.)

This is very similar to the flipping coin problem. While the movement of the coin can be seen as absolutely deterministic, still the outcome can not be predicted in a reasonable way, because small alterations in the initial state can grow large enough to change the result.

(Not to mention that the required "very high precision", if for example we talk about spatial position, can easily be much less than the size of an atom, which is beyond the limits of applicability of classical mechanics. That it, the result may radically change if you change the initial position by a fraction of atom size; but you can not require a classical mechanics initial state to be specified to that precision.)

This is very similar to the flipping coin problem. While the movement of the coin can be seen as absolutely deterministic, still the outcome can not be predicted in a reasonable way, because small alterations in the initial state can grow large enough to change the result.