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Since a small change in the initial state can result in completely different behavior, doesn't this mean our numerical approximations have to be exact to model the system? In particular, won't floating-point rounding reflect a small change in the state and therefore model a completely different system?


How to estimate the accuracy of a given approximation algorithm on the double pendulum system?


If a small change in input result in wild changes in output, are there some methods of numerical approximation that fits the double pendulum better that we should use? Or is this again a tradeoff situation?


In my past experiences, modeling physical motion with kinematic equations is quite difficult (just plugging numbers into the formulas doesn't always get us the right results). How would we find a good numerical approximation for a system like this?


What kind of numerical approximation algorithms can we use here?


I think the accuracy may not very important in this scenario. Because even small changes can influence whole system and people may not care too much in such a scene.


How much approximation can we use here? What if we do not know how much difference would make to the whole system for a minor approximation in advance?


What kind of approximation algorithms would we use? How would we evaluate how good such an algorithm is?