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> perfect prediction is in principle impossible. “You can’t include everything in the universe in your calculation,” The issue is even worse. Even if you just
by rocqua 8y ago
> perfect prediction is in principle impossible. “You can’t include everything in the universe in your calculation,”
The issue is even worse. Even if you just include the Earth, Moon and sun, the actual orbit is impossible to calculate. That is, we do not have an exact solution, and approximate solutions diverge quickly, and require a lot more computation to remain accurate for a bit longer. This is all related to the fact that it is very hard to establish whether a 3 body system of bodies is even 'stable'. That is, whether one of the bodies reaches escape velocity and just shoots off.
- garmaine 8y agoThis isn’t true. We have very reliable libraries for simulating N-body problems to sufficient detail to know the answers to such questions with great certainty. Not having an “exact solution” Does not bound whether solutions exist or whether they are computable, or how easy they are to compute. In this case, quite easy.
- teraflop 8y agoIn principle, orbits in a 3-body system are unstable when extrapolated arbitrarily far into the future. But in the case of the Earth-Moon system, it's easy to find orbits where the "instability" grows on a timescale of years or decades, and therefore can be easily dealt with by active course corrections.
- troymc 8y agoThe belief that there's no exact solution to the three-body problem is common but wrong. There is one, and it was found about a hundred years ago. The history of the problem, and its solution, are all described in the book Poincare and the Three Body Problem, published by the American Mathematical Society. It includes a section about potential reasons for the pervasive belief that no solution exists.
- rocqua 8y ago" A complete solution for a particular three-body problem would provide the positions for all three particles for all time, given three initial positions and initial velocities. In general, no closed-form solution for such a problem exists, and the time evolution of the system is believed to be chaotic. " from wikipedia [1]. Not a fool-proof source, but it seems to disagree with you. It does seem the case that whether the system is chaotic isn't yet determined. [1] https://en.wikipedia.org/wiki/Three-body_problem#Gravitational_systems https://en.wikipedia.org/wiki/Three-body_problem#Gravitation...
- privong 8y agoThere is a series solution to the 3-body problem, which can in principle be computed to an arbitrary level of precision. So while there's no closed-form solution (as the wikipedia article notes), there is a solution. However, my understanding is that this series solution converges slowly enough as to not be useful in practice.
- rocqua 8y agoAh, that is interesting. I figure Numerical ODE solvers might be faster. But there is philosophical value in a series that is proven to converge for any time t. Do you happen to recall whether it is uniform convergence?
- privong 8y agoI don't have more than a cursory knowledge of the solution, but the Three-body problem wikipedia page should have enough info to get you set in the right direction: https://en.wikipedia.org/wiki/Three-body_problem#Sundman's_theorem https://en.wikipedia.org/wiki/Three-body_problem#Sundman's_t... It says that the series should converge for all t, except for systems which begin with zero angular momentum. It additionally states that: > in 1930, David Beloriszky calculated that if Sundman’s series were to be used for astronomical observations, then the computations would involve at least 10^8000000 terms