“The Most Celebrated of all Dynamical Problems” History and Details to the Restricted Three Body Problem David Goodman 12/16/03 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.

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

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

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

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

Happy Birthday King Oscar!!! The Occasion: For his 60 th 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

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

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

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

The Players

Problem with Oscar’s Jury: Language Barrier Distance Rivalry

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

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

The Players Leffler Weierstrass Hermite

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

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

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

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

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

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

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

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

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

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

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

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

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

The Solution Definitions – Represents the three particles – Represents the mass of each –Distance –

The Solution The equations of motion –Based on Newton’s law of gravitation

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

The Solution Equations in the Hamiltonian form:

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

The Solution Integrate twice and are constants of integration

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:

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:

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:

The Solution and

The Solution and

The Solution and Then add the two together to get

The Solution Permute cyclically the variable and integrate to obtain

The Solution Consider Then

The Solution Multiply by and sum to get integrate

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

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

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

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

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

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

The Solution Periodic solutions –Poincaré’s theory –Depend on initial conditions

The Solution What is a periodic solution? –A solution is periodic with period if when is a linear variable and is an angular variable

The Solution We’ll focus on this the most concise of his mathematical solutions Trigonometric series approach –Used trig series of the form

The Solution Tried to find a general solution for the system of linear differential equations coefficients are periodic functions of with period

The Solution Began with

The Solution Next

The Solution Then a linear combination of the original solutions Constant

The Solution Let be the root of the eigenvalue equation

The Solution Then Constant such that and Then we can expandas trig series

The Solution Finally… –Poincaré wrote his final solution to the system of differential equations as

And it Goes on… Lemmas, theorems,corollaries invariant integrals, proofs I’m starting feel like the jury who studied the original 198 pages The rest of Poincaré’s solution was an attempt to generalize the solution for the n body problem

To conclude Study the three body problem to hone your mathematical and dynamical skills Kronecker hated everybody Poincaré was a nice guy with a good solution

Works Cited Barrow-Green, June. Poincaré and the Three Body Problem. History of Mathematics, Vol. 11. American Mathematical Society, Goldstein, Herbert; Poole, Safko. Classical Mechanics. 3 rd ed. Addison Wesley, Szebehely, Victor. Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press, Whittaker, E.T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, 1965.