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Chaos, Planetary Motion and the n -Body Problem

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Planetary Motion What paths do planets follow? Why? Will they stay that way forever?

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E pur si muove! Aristarchos (~310 B.C) suggested a heliocentric universe. Galileo Galilei (~1600 B. C) established from observation that the Earth did orbit the Sun Galileo still believed that orbits were circular. However, a contemporary had other ideas...

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Johannes Kepler ( ) Observed that orbits tend to follow elliptical paths with the Sun at one focus Found by astronomical observation. Scientists were still unclear as to the reason why.

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Sir Isaac Newton ( ) Formulated three laws of motion From these laws, derived the Law of Universal Gravitation

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Sir Isaac Newton ( ) Formulated three laws of motion From these laws, derived the Law of Universal Gravitation This is the only equation in this presentation, I promise!

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What does that all mean? Scientists now had equations that govern the motion of planets Johann Bernoulli showed that when there were two planets in a stable system, the orbit was elliptical. Kepler was right! Other unstable orbits were also possible: Hyperbolas, Parabolas (comets!) What about more than two planets?

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What does that all mean? Scientists now had equations that govern the motion of planets Johann Bernoulli showed that when there were two planets in a stable system, the orbit was elliptical. Kepler was right! Other unstable orbits were also possible: Hyperbolas, Parabolas (comets!) What about more than two planets?

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What does that all mean? Scientists now had equations that govern the motion of planets Johann Bernoulli showed that when there were two bodies in a stable system, the orbit was elliptical. Kepler was right! Other unstable orbits were also possible: Hyperbolas, Parabolas (comets!) What about more than two bodies?

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Some Simple Solutions Straight Line Paths Elliptical Paths Circular Orbits

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King Oscar II (1829 – 1907) King Oscar II of Sweden was concerned about the future of the Solar System Offered a prize if someone could find a solution for the general n -body problem This prompted interest from one of the mathematical superpowers of the day…

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Henri Poincaré (1854 – 1912) The greatest applied mathematician of his time Soon realised that he may have bitten off somewhat more than he could chew Even the 3-body problem turns out to be far more tricky than previously imagined

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Henri Poincaré (1854 – 1912) The greatest applied mathematician of his time Soon realised that he may have bitten off somewhat more than he could chew Even the 3-body problem turns out to be far more tricky than previously imagined

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Small changes ≠ Small Effects The problem is harder than anyone realised Poincaré was forced to develop a mathematical theory of ’small bumps’, known as asymptotic theory Mathematicians could now look at what happens when solutions are given a small nudge

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CHAOS! RUN! Slight changes in starting behaviour can lead to significant changes in long-term behaviour Even if we know the position of every body to high precision, we cannot guarantee that a solution will hold for all time Poincaré used these ideas to show that there is no practical solution to the n -body problem

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Where does that leave us? In 1912, Sundman found a solution that is valid for almost all cases; however, it is not very useful for computations It turns out that the solar system may not be stable, but it is predictable for very long timescales – we are probably alright! Asymptotic theory was applied to a large number of mathematical disciplines

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Fluid Dynamics

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Modern Physics

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And Much, Much More!

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So what was the point? Predicting two bodies is simple, but when more bodies are introduced, chaos ensues Problem is sensitive to initial configuration, making predictions difficult, although solar system is stable for the foreseeable future Asymptotics became widely recognised as an important technique for applied mathematicians

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So what was the point? Predicting two bodies is simple, but when more bodies are introduced, chaos ensues Problem is sensitive to initial configuration, making predictions difficult, although solar system is stable for the foreseeable future Asymptotics became widely recognised as an important technique for applied mathematicians Finally, orbits look cool

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What does a mathematician do all day?

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