Organising Centres in the Semi-global Analysis of Dynamical Systems
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Selection results: MathMods Selection results for the edition of the MathMods programme are out now! Student Information System A gateway to various services, like transcripts of records, fee tracking and much more. Check your Mail Click here to check your mathmods. The problem was stated essentially as follows: Will the planets of the solar system continue forever in much the same arrangement as they do at present? Or could something dramatic happen, such as a planet being flung out of the solar system entirely or colliding with the Sun? Mathematicians already knew that considerable difficulties arise in answering any such questions as soon as the number of bodies involved exceeds two.
Dynamical systems theory and chaos
For two bodies moving under Newtonian gravitation, it is possible to solve the differential equation and deduce an exact formula for their motion: they move in ellipses about their mutual centre of gravity. It turned out that even the restricted three-body problem was still too difficult to be solved.
By ingenious geometric arguments, he showed that planetary orbits in the restricted three-body problem are too complicated to be describable by any explicit formula.
Suppose one knows some solution path and wants to find out how nearby solution paths behave. Imagine a surface that slices through the known path. Nearby paths will also cross this surface and may eventually return to it. Even when there is no hint of randomness in the equations, there can be genuine elements of randomness in the solutions. The Russian school of dynamicists under Andrey Kolmogorov and Vladimir Arnold developed similar ideas at much the same time.
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Dynamical systems theory - Wikipedia
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