Determinism, chaos and chance Henk Broer Johann Bernoulli Institute for Mathematics and Computer Science University of Groningen Summary i. Stability of solar system ii. Chaos versus chance iii. ... Email: [email protected] URL: http://www.math.rug.nl/˜broer Heroes - Kepler and Galileo - Newton and Laplace - Poincaré and Kolmogorov Kepler and Galileo Iohannes Kepler Galileo Galilei (1571-1530) (1564-1642) Observing and thinking Newton and Laplace Sir Isaac Newton Pierre-Simon Laplace (1642-1727) (1749-1827) Solar system deterministic: then how about perpetual stability? Kepler I & II Kepler I: Elliptic orbit with Sun in a focal point Kepler II: Constant sectorial velocity Circular orbit y r F 0 x km Circular orbit in central force field F = − r2 er 2π x(t) cos T t r(t)= = R 2π y(t) sin T t Problem: what is relationship between R and T ? Centripetal acceleration y y v a 0 x 0 x Velocity and (centripetal) accelaration Kepler III from Newton’s laws Centripetal acceleration 2 2π x¨(t) 2π cos T t a(t) = = −R 2π y¨(t) T sin T t 2 2π = −R er T Combining Newton’s laws km m a = F = − er r2 find 4π2 T 2 = R3 (Kepler III) k Galilean moons of Jupiter Revolutions in about Io: 2 days, Europa: 4 days, Ganymedes: 1 week, Callisto: 2 weeks Does Kepler III hold ? Checked by Flamsteed: YES R.S. Westfall, Never at Rest, Cambridge University Press 1981 Scholium: chaos? • Universal gravitation: no longer them nice ellipses! but perturbation theory • Poincaré and the three body problem: homoclinic tangle Henri Poincaré (1854-1912) and his tangle June Barrow-Green, Poincaré and the Three Body Problem. History of Mathematics, Volume 11 American Mathematical Society / London Mathematical Society 1997 Hénon-Heiles 1964: a toy model Coupled oscillators ∂V x′′ = − ∂x ∂V y′′ = − ∂y 1 2 2 2 2 3 potential energy V (x,y)= 2(x + y +2x y − 3y ) 1 ′ 2 ′ 2 total energy E = 2 (x ) +(y ) + V (x,y): a conserved quantity Setting x′ := u and y′ := v phase-space R4 = {x,y,u,v} Three sphere S3 ⊂ R4 Energy hypersurface 2 2 2 2 2 2 3 S3 x + y + u + v +2x y − 3y = E ≈ sphere Geometry of S3 ≈ R3 ∪ {∞} 2–dimensional torus T2 S3 ≈ union of two solid tori glued along common boundary T2 ... Three sphere S3, ctd. 4 2 0 -2 -4 2 0 -2 -4 -2 0 2 4 Seiffert foliation of S3 Taking Poincaré section transverse to such 2–tori ... Hénon-Heiles II qualitative picture of the dynamics Energy E =0.005 (left) and E =0.010 (right) mainly (multi-) periodic ≡ stable Hénon-Heiles III E =0.012 a lot of ‘chaos’ too ... The swing Stroboscopic pictures of the swing y x Without (left) and with dissipation (right) Conservative chaos and a Hénon-like strange attractor Invariant measures, ergodicity Andrei N. Kolmogorov Yakov G. Sinai (1903-1987) (1935- ) • Poincaré recurrence theorem • probability, measure, ergodicity • also for dissipative systems Moreover ... • Open problems in mathematics: - Are (physical) measures ergodic? - Relation to geometry of unstable manifold and homoclinic tangle? • Jacques Laskar (Observatoire de Paris): (inner) solar system chaotic, to be noticed in about 100 000 000 yrs J. Laskar and M. Gastineau, Existence of collisional trajectories of Mercury, Mars and Venus with the Earth. Nature Letters 459|11 June 2009|doi:10.1038/nature08096 V.I. Arnold and A. Avez, Probèmes Ergodiques de la Mécanique classique, Gauthier-Villars, 1967; Ergodic problems of classical mechanics, Benjamin 1968 J. Palis and F. Takens, Hyperbolicity & Sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics 35, Cambridge University Press 1993 Edward Lorenz 1963 Edward Norton Lorenz (1917-2008) Chaos awakening E.N. Lorenz, Deterministic nonperiodic flow, J. Atmosph. Sci. 20 (1963), 130-141 H.W. Broer and F. Takens, Dynamical Systems and Chaos. Applied Mathematical Sciences 172, Springer, 2011 Lorenz system and attractor ′ x = σy − σx ′ y = rx − y − xz ′ z = −bz + xy, with σ = 10, b =8/3 and r = 28 Nature of turbulence David Ruelle Floris Takens (1935- ) (1940-2010) Turbulence: multiperiodicity or chaos? Memento Heisenberg and Lamb ... Scholium: chaos versus chance • Magnetic pendulum / compare dice • Boltzmann, Gibbs: Statistical physics • Life itself • Quantum physics.