“The Most Celebrated of all Dynamical Problems”

1. This presentation will probably involve audience discussion, which will create action items. Use PowerPoint to keep track of these action items during your presentation • In Slide Show, click on the right mouse button • Select “Meeting Minder” • Select the “Action Items” tab • Type in action items as they come up • Click OK to dismiss this box • This will automatically create an Action Item slide at the end of your presentation with your points entered. “The Most Celebrated of all Dynamical Problems” History and Details to the Restricted Three Body Problem David Goodman 12/16/03

2. History of the Three Body Problem The Occasion The Players The Contest The Champion

3. Details and Solution of the Restricted Three Body Problem The Problem The Solution

4. King Oscar

5. King Oscar King Oscar: • Joined the Navy at age 11, which could have peaked his interest in math and physics • Studied mathematics at the University of Uppsala • Crowned king of Norway in 1872

6. King Oscar • Distinguished writer and musical amateur • Proved to be a generous friend of learning, and encouraged the development of education throughout his reign • Provided financial support for the founding of Acta Mathematica

7. Happy Birthday King Oscar!!! The Occasion: • For his 60th birthday, a mathematics competition was to be held • Oscar’s Idea or Mitag-Leffler’s Idea? • Was to be judged by an international jury of leading mathematicians

8. The Players Gösta Mittag-Leffler: • A professor of pure mathematics at Stockholm Höfkola • Founder of Acta Mathematica • Studied under Hermite, Schering, and Weierstrass

9. The Players Gösta Mittag-Leffler: • Arranged all of the details of the competition • Made all the necessary contacts to assemble the jury • Could not quite fulfill Oscar’s requirements for the contest

10. The Players Oscar’s requested Jury: • Leffler, Weierstrass, Hermite, Cayley or Sylvester, Brioschi or Tschebyschev • This jury represented each part of the world

11. The Players

12. The Players Problem with Oscar’s Jury: • Language Barrier • Distance • Rivalry

13. The Players The Chosen Jury: • Hermite, Weierstrass and Mittag-Leffler • All three were not rivals, but had great respect for each other

14. The Players “You have made a mistake Monsieur, you should of taken the courses of Weierstrass in Berlin. He is the master of us all.” –Hermite to Leffler • All three were not rivals, but had great respect for each other

15. The Players Leffler Weierstrass Hermite

16. The Players Kronecker: • Incensed at the fact that he was not chosen for jury • In reality, probably, more upset about Weierstrass being chosen • Publicly criticized the contest as a vehicle to advertise Acta

17. The Players The Contestants: • Poincaré • Chose the 3 body problem • Student of Hermite • Paul Appell • Professor of Rational Mechanics in Sorbonne • Student of Hermite • Chose his own topic • Guy de Longchamps • Arrogantly complained to Hermite because he did not win

18. The Players The Contestants: • Jean Escary • Professor at the military school of La Fléche • Cyrus Legg • Part of a “band of indefatigable angle trisectors”

19. The Contest • Mathematical contests were held in order to find solutions to mathematical problems • What a better way to celebrate, a mathematician’s birthday, the King, than to hold a contest • Contest was announced in both German and French in Acta, in English in Nature, and several languages in other journals

20. The Contest • There was a prize to be given of 2500 crowns (which is half of a full professor’s salary) • This particular contest was concerned with four problems • The well known n body problem • A detailed analysis of Fuch of differential equations • Investigation of first order nonlinear differential equations • The study of algebraic relations connecting Poincaré Fuchsian functions with the same automorphism group

21. The Champion • Poincaré • He was unanimously chosen by the jury • His paper consisted of 158 pages • The importance of his work was obvious • The jury had a difficult time understanding his mathematics

22. The Champion “It must be acknowledged, that in this work, as in almost all his researches, Poincaré shows the way and gives the signs, but leaves much to be done to fill the gaps and complete his work. Picard has often asked him for enlightenment and explanations and very important points in his articles in the Comptes Rendes, without being able to obtain anything, except the statement: ‘It is so, it is like that’, so that he seems like a seer to whom truths appear in a bright light, but mostly to him alone…”.- Hermite

23. The Champion • Leffler asked for clarification several times • Poincaré responded with 93 pages of notes

24. The Problem • Poincaré produced a solution to a modification of a generalized n body problem known today as the restricted 3 body problem • The restricted 3 body problem has immediate application insofar as the stability of the solar system

25. The Problem • “I consider three masses, the first very large, the second small, but finite, and the third infinitely small: I assume that the first two describe a circle around the common center of gravity, and the third moves in the plane of the circles.” -Poincaré

26. The Problem • “An example would be the case of a small planet perturbed by Jupiter if the eccentricity of Jupiter and the inclination of the orbits are disregarded.” -Poincaré

27. The Solution • “It’s a classic three body problem, it can’t be solved.”

28. The Solution • “It’s a classic three body problem, it can’t be solved.” • It can, however, be approximated!

29. The Solution • Definitions • Represents the three particles • Represents the mass of each • Distance

30. The Solution • The equations of motion • Based on Newton’s law of gravitation

31. The Solution • The task is to reduce the order of the system of equations • Choose • Force between and becomes: • Potential energy of the entire system

32. The Solution • Equations in the Hamiltonian form:

33. The Solution • We now have a set of 18 first order differential equations (that’s a lot) • We shall now attempt to reduce them • Multiply original equations of motion by

34. The Solution • Integrate twice • and are constants of integration

35. The Solution • Since the integral is a constant the motion of the center of mass is either stationary or moving at constant velocity. • How about some confusion? Multiply:

36. The Solution • Since the integral is a constant the motion of the center of mass is either stationary or moving at constant velocity. • How about some confusion? Multiply:

37. The Solution • Since the integral is a constant the motion of the center of mass is either stationary or moving at constant velocity. • How about some confusion? Multiply:

38. The Solution • and

39. The Solution • and

40. The Solution • and • Then add the two together to get

41. The Solution • Permute cyclically the variable and integrate to obtain

42. The Solution • Consider • Then

43. The Solution • Multiply by and sum to get • integrate

44. The Solution • The final reduction is the elimination of the time variable by using a dependent variable as an independent variable • Then a reduction through elimination of the nodes

45. The Solution “Damn it Jim, I’m a doctor, not a mathematician!”

46. The Solution • Now our system of equation is reduced from an order of 18 to an order of 6 • Let’s apply it to the restricted three body problem and attempt a solution

47. The Solution • There are several different avenues to follow at this point • Particular solutions • Series solutions • Periodic solutions

48. The Solution • Particular solutions • Impose geometric symmetries upon the system • Examples in Goldstein • Lagrange used collinear and equilateral triangle configurations

49. The Solution • Series solutions • Much work done in series solutions • Problem was with convergence and thus stability • Converged, but not fast enough

50. The Solution • Periodic solutions • Poincaré’s theory • Depend on initial conditions