Astro 201 Sept 9 2010 Do on line practice quiz 2 by 9 14 Turn in HW 2 in box in front of the room Reading Hester Chapter 4 Today String Theory Determinism v Chaos Fractals Light STRING THEORY everything is ultimately made of strings sub sub sub atomic particles How big are strings Smaller than a Planck length which is about 10 33 centimeters or about a millionth of a billionth of a billionth of a billionth of a centimeter Strings vibrate Closed string Open string In many string theories the Universe is 10 dimensional with the extra dimensions COMPACTIFIED All HS math geeks read a book called FLATLAND A Romance of Many Dimensions by Edwin A Abbott 1884 FLATLAND a first person account of life of a 2 dimensional society a radical named Arthur Square figures out that space is really 3 dimensional So what the string theorists say is that in ordinary life we think we live in 3 dimensions and we have to think of ways to detect the other 7 The extra dimensions in string theory are Calabi Yau figures Knitted Calabi Yau Figures Kepler Empirical description of the motion of the planets Newton Law of Gravity Developed Calculus derived orbits of the planets Solved the Two body problem Sun one planet Couldn t solve the Three body problem Mechanical Universe In Newtonian physics objects move in perfectly determined ways Orrery Mechanical models of planetary motions in the solar system But is the real solar system accurately described by an orrery We may regard the present state of the universe as the effect of its past and the cause of its future An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it if this intellect were vast enough to submit the data to analysis could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes Marquis Pierre Simon de Laplace 1749 1827 The Equations Many thanks To Scott Tremaine s Notes from April 2006 King Oscar II of Sweden 1829 1907 Prize How stable is the universe Jules Henri Poincar 1854 1912 Sun large plus one planet circular orbit Stable Added 3rd body Strange behavior Not periodic Modern approach Solve many body problem with computer calculations Take a distribution of mass Figure out the gravitational force on each part F ma gives you the acceleration on each part Compute velocity of each part Move the parts a little Repeat Kuiper belt objects Plutinos 3 2 Centaurs comets as of March 8 2006 Minor Planet Center Sensitivity to Initial Conditions A very small cause which escapes our notice determines a considerable effect that we cannot fail to see and then we say that the effect is due to chance If we knew exactly the laws of nature and the situation of the universe at the initial moment we could predict exactly the situation of the same universe at a succeeding moment But even if it were the case that the natural laws had no longer any secret for us we could still know the situation approximately If that enabled us to predict the succeeding situation with the same approximation that is all we require and we should say that the phenomenon had been predicted that it is governed by the laws But is not always so it may happen that small differences in the initial conditions produce very great ones in the final phenomena A small error in the former will produce an enormous error in the latter Prediction becomes impossible Poincar Can we predict the motion of a single planet a billion years from now Laplace and Newton Yes Poincare No Lorenz 1963 Butterfly Effect If a butterfly flaps its wings in Brazil does it result in a tornado in Kansas Two kinds of dynamical systems Regular highly predictable wellbehaved e g baseball golf simple pendulum all problems in mechanics textbooks planetary orbits on short timescales Chaotic difficult to predict erratic appears regular on timescales short compared to Liapunov time e g roulette dice pinball weather billiards double pendulum The Solar System Double Pendulum a chaotic system Consequences of chaos Positions of planets are not predictable on timescales longer than 100 Myr the solar system is a poor example of a deterministic universe The solar system is Chaotic Fractals Geometric forms Define by a recursive rule Same on all scales Benoit Mandelbrot Serpinski Triangle Von Koch Snowflake Fractals are SelfSimilar same when you zoom in The Julia Set Gaston Julia French mathematician The Mandelbrot Set Fractals in Nature