Michel Hénon A playful and simplifying spirit

Transcription

1 Michel Hénon A playful and simplifying spirit Jean Marc Petit Not a Science talk! More of a story of how I see Michel.

2 My first meeting DEA «Turbulence et Systèmes Dynamiques» class Not a class on dynamical systems Rather «practical» programming lecture

3 My first meeting Most programs, including mine: Michel's view A real mess Structured programming

4 My first meeting

5 My first meeting Global vision, «top down» approach Extract the hard point of the problem Structured analysis of the problem Write a clear documentation Rigorous methodology Michel's main strength : Stick to these rules in all cases

6 It's about choice

7 Saturn's rings

8 Saturn's rings Voyager data Saturn's rings way more complex than previously thought Simplified model with essence of the problem 2 dimensional problem Succession of binary interactions 2 physical effects: gravitation and collisions Looking at radial structure, so keep only radial coordinate Particles on almost circular orbits

9 Saturn's rings Determine effect of gravitational interaction between 2 particles in orbit around Saturn at large distance Involves polynomial and trigonometric series expansions Tedious manual work

10 Saturn's rings

11 Saturn's rings So Michel wrote a symbolic computation system in FORTH (using a self writen text editor)

12 Saturn's rings

13 Saturn's rings Now, have a numerical model of rings evolution Not enough, need analytical understanding Allow testing of computer model Write diffusion equation

14 Saturn's rings Solve for self similar asymptotic solution Width (X0) as (T T0)1/3 Height as (T T0)-1/3

15 Saturn's rings to inclined billiard

16 Saturn's rings to inclined billiard

17 Saturn's rings to inclined billiard One can define a symbolic dynamics D of symbols dj

18 Collisions and fragmentation Modelling of fragmentation process in hyper velocity collisions in asteroid belt Usually assume size distribution of fragments No physics Don't know if fragments reconstruct parent body Geometric model of fragments Make sure the fragments really enter the parent body Do we recover a well defined size distribution?

19 Collisions and fragmentation

20 Collisions and fragmentation

21 Collisions and fragmentation

22 Collisions and fragmentation

23 Collisions and fragmentation

24 Collisions and fragmentation

25 Collisions and fragmentation

26 Collisions and fragmentation

27 Collisions and fragmentation Define number of fragments f(n,r) with limit f(r) and probability of a fragment of size r at trial n Relate to surface of previous fragments and empty volume

28 Collisions and fragmentation Assumption: scale-free problem power-law Solve for exponant (a is dimensionality):

29 Symplectic integrators and rotations Long term integrations of Hamiltonian systems require to preserve energy and other integrals Use of symplectic integrator help for this But rotations ruins this behaviour

30 Symplectic integrators and rotations Solve the problem by using 3 shears End of the problem Not for Michel

31 Symplectic integrators and rotations Approximate values of numbers in IEEE754 Solve for:

32 Symplectic integrators and rotations

33 Symplectic integrators and rotations In double precision, uses number theory on diophantine equation Number of solutions of sub class, and solutions

34 Symplectic integrators and rotations It works!

35 Michel, you'll stay in my mind and my heart for ever.