Chaos, PlanetaryMotion and the nBody Problem混沌行星的運動和N體問題

1、chaos, planetary motion and the n-body problemplanetary motionwhat paths do planets follow? why?will they stay that way forever?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 sungalileo stil

2、l believed that orbits were circular. however, a contemporary had other ideas.johannes kepler (1571 - 1630)observed that orbits tend to follow elliptical paths with the sun at one focusfound by astronomical observation. scientists were still unclear as to the reason why. sir isaac newton (1642 - 174

3、2)formulated three laws of motionfrom these laws, derived the law of universal gravitation sir isaac newton (1642 - 1742)formulated three laws of motionfrom these laws, derived the law of universal gravitation this is the only equation in this presentation,i promise!what does that all mean?scientist

4、s now had equations that govern the motion of planetsjohann 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?what does that all me

5、an?scientists now had equations that govern the motion of planetsjohann 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?what does

6、 that all mean?scientists now had equations that govern the motion of planetsjohann 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

7、?some simple solutionsstraight line pathselliptical pathscircular orbitsking oscar ii (1829 1907)king oscar ii of sweden was concerned about the future of the solar systemoffered a prize if someone could find a solution for the general n-body problemthis prompted interest from one of the mathematica

8、l superpowers of the dayhenri poincar (1854 1912)the greatest applied mathematician of his timesoon realised that he may have bitten off somewhat more than he could cheweven the 3-body problem turns out to be far more tricky than previously imaginedhenri poincar (1854 1912)the greatest applied mathe

9、matician of his timesoon realised that he may have bitten off somewhat more than he could cheweven the 3-body problem turns out to be far more tricky than previously imagined/charlie/3body//charlie/3body/small changes small effectsthe problem is harder

10、 than anyone realisedpoincar was forced to develop a mathematical theory of small bumps, known as asymptotic theorymathematicians could now look at what happens when solutions are given a small nudgehttp:/ run!slight changes in starting behaviour can lead to significant changes in long-term behaviou

11、reven if we know the position of every body to high precision, we cannot guarantee that a solution will hold for all timepoincar used these ideas to show that there is no practical solution to the n-body problemwhere does that leave us?in 1912, sundman found a solution that is valid for almost all c

12、ases; however, it is not very useful for computationsit 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 disciplinesfluid dynamicsmodern physicsand much, much more

13、!so what was the point?predicting two bodies is simple, but when more bodies are introduced, chaos ensuesproblem is sensitive to initial configuration, making predictions difficult, although solar system is stable for the foreseeable futureasymptotics became widely recognised as an important technique for applied mathematiciansso what was the point?predicting two bodies is simple, but when more bodies are introduced, chaos ensuesproblem is sensitive to initial configuration, making predictions difficult,