Determinism, chaos and chance

Henk Broer

Johann Bernoulli Institute for Mathematics and Computer Science University of 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 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 Invariant measures,

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 ): (inner) solar system chaotic, to be noticed in about 100 000 000 yrs

J. Laskar and M. Gastineau, Existence of collisional trajectories of , and with the . 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 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