“The Most Celebrated of all Dynamical Problems” History and Details to the Restricted Three Body Problem David Goodman 12/16/03 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 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 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 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 The Players Problem with Oscar’s Jury: Language Barrier Distance Rivalry The Players The Chosen Jury: Hermite, Weierstrass and MittagLeffler 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 – Pi Represents the three particles – mi Represents the mass of each – Distance Pi Pj rij – i 1,2,3 The Solution The equations of motion – Based on Newton’s law of gravitation d 2 q1i q2i q1i 2 q3i q1i 2 k m2 k m3 2 3 dt r12 r133 d 2 q2i q1i q2i 2 q3i q2i 2 k m2 k m3 2 3 dt r12 r233 d 2 q3i q1i q3i 2 q2i q3i 2 k m2 k m3 2 3 dt r13 r233 The Solution The task is to reduce the order of the system of equations Choose k 2 1 Force between i and j becomes: mi m j rij2 Potential energy of the entire system m2 m3 m3m1 m1m2 V r23 r31 r12 The Solution pij mi dqij dt 3 pij2 i , j 1 2mi H V Equations in the Hamiltonian form: H dt pij dqij H dt qij dpij 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 2 3 d 2 qij d qij mi 2 mi 0 2 dt dt i 1 The Solution Integrate twice 3 dt m d qij i i 1 2 dt Aj and B j integration 2 3 mi qij Aj t B j i i are constants of 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: d 2 q11 q12 dt 2 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: 2 d 2 q11 d q12 q12 q22 dt 2 dt2 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: d 2 q11 d 2 q12 q12 q22 2 dt dt2 d 2 q13 q32 dt2 The Solution and d 2 q21 q11 dt 2 The Solution and d 2 q21 q11 dt 2 d 2 q22 q21 dt2 The Solution and d 2 q21 q11 dt 2 d 2 q22 q21 dt2 d 2 q23 q31 dt2 Then add the two together to get d 2 qi 2 3 d 2 qi1 mi qi1 mi qi 2 0 2 2 dt dt i 1 i 1 3 The Solution Permute cyclically the variable and integrate to obtain dqi 2 dqi 3 mi qi 2 qi 3 C1 dt dt i 1 3 dqi 3 dqi1 mi qi 3 qi1 C2 dt dt i 1 3 dqi1 dqi 2 mi qi1 qi 2 C3 dt dt i 1 3 The Solution Consider qkj qij 1 qij rik rik3 Then mi d 2 qij dt 2 V qij The Solution Multiply by 3 p i , j 1 d 2 qij ij dt2 dqij dt dV dt integrate 3 pij2 2m i , j 1 i V C and sum to get 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 x1 1 t ,...,xn n t is periodic with period h if when x is a linear variable i t h i t and x i is an angular variable i t h i t 2k k integer The Solution We’ll focus on this the most concise of his mathematical solutions Trigonometric series approach – Used trig series of the form f x Ao A1 cos x ... An cosnx ... B1 sin x ... Bn sin nx ... The Solution Tried to find a general solution for the system of linear differential equations dxi i.1 x1 ... i.n xn dt coefficients i.k are periodic functions of t with period 2 n 2 The Solution Began with x1 i.1 t ,...,xn i.n t i 1,...,n The Solution Next x1 i.1 t 2 ,...,xn i.n t 2 i 1,...,n The Solution Then a linear combination of the original solutions xi.k t 2 Ai.1 1.k t ,..., Ai.n xn.k t A Constant The Solution Let S1 be the root of the eigenvalue equation A1.1 S A2.1 ... An.1 A1.2 ... A1.n A2.2 S ... ... ... A2.n ... An.2 ... An.n S 0 The Solution Then Constant Bk such that 1.i t 2 S11.i t n and 1.i t Bk k .i k 1 Then we can expand S11.i t as trig series 1.i The Solution Finally… – Poincaré wrote his final solution to the system of differential equations as xi e 1.i t 1t 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, 1997. Goldstein, Herbert; Poole, Safko. Classical Mechanics. 3rd ed. Addison Wesley, 2002. Szebehely, Victor. Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press, 1967. Whittaker, E.T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, 1965.