N body problems mostly don't admit any algebraic solutions although they can be solved numerically at least in principle. Determinism basically is the claim that given any system and its initial conditions one could in theory always calculate all the information about that system in any given time. This leads to the concept of deterministic chaos, systems that are very sensitive to initial conditions tend to be for all practical purposes unpredictable and therfore their state undeterminable. But theoretically given any amount of computing power can you determine the state of any classical system, or is classical physics not a completely deterministic theory?
This is not about non uniqueness or some artificial examples. The point is for example the problem of the stability of the solar system. Given a gravitating n body system can you find its state at any time t. Can this be proven to be undecidable.