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The Most Celebrated of all Dynamical Problems

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• History and Details to the Restricted Three Body
  Problem
• David Goodman
• 12/16/03

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History of the Three Body Problem
The Occasion The Players The Contest The Champion
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Details and Solution of the Restricted Three
Body Problem
The Problem The Solution
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King Oscar
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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

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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

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Happy Birthday King Oscar!!!

• The Occasion
• For his 60th birthday, a mathematics competition
  was to be held
• Oscars Idea or Mitag-Lefflers Idea?
• Was to be judged by an international jury of
  leading mathematicians

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The Players

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

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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 Oscars requirements for
  the contest

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The Players

• Oscars requested Jury
• Leffler, Weierstrass, Hermite, Cayley or
  Sylvester, Brioschi or Tschebyschev
• This jury represented each part of the world

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The Players
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The Players

• Problem with Oscars Jury
• Language Barrier
• Distance
• Rivalry

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The Players

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

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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

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The Players
Leffler Weierstrass
Hermite
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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

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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

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The Players

• The Contestants
• Jean Escary
• Professor at the military school of La Fléche
• Cyrus Legg
• Part of a band of indefatigable angle
  trisectors

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The Contest

• Mathematical contests were held in order to find
  solutions to mathematical problems
• What a better way to celebrate, a mathematicians
  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

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The Contest

• There was a prize to be given of 2500 crowns
  (which is half of a full professors 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

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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

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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

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The Champion

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

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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

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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

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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

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The Solution

• Its a classic three body problem, it cant be
  solved.

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The Solution

• Its a classic three body problem, it cant be
  solved.
• It can, however, be approximated!

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The Solution

• Definitions
• Represents the three particles
• Represents the mass of each
• Distance

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The Solution

• The equations of motion
• Based on Newtons law of gravitation

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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

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The Solution

• Equations in the Hamiltonian form

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The Solution

• We now have a set of 18 first order differential
  equations (thats a lot)
• We shall now attempt to reduce them
• Multiply original equations of motion by

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The Solution

• Integrate twice
• and are constants of integration

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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

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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

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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

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The Solution

• and

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The Solution

• and

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The Solution

• and
• Then add the two together to get

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The Solution

• Permute cyclically the variable and integrate to
  obtain

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The Solution

• Consider
• Then

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The Solution

• Multiply by and sum to get
• integrate

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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

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The Solution

• Damn it Jim, Im a doctor, not a mathematician!

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The Solution

• Now our system of equation is reduced from an
  order of 18 to an order of 6
• Lets apply it to the restricted three body
  problem and attempt a solution

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The Solution

• There are several different avenues to follow at
  this point
• Particular solutions
• Series solutions
• Periodic solutions

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The Solution

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

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The Solution

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

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The Solution

• Periodic solutions
• Poincarés theory
• Depend on initial conditions

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The Solution

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

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The Solution

• Well focus on this the most concise of his
  mathematical solutions
• Trigonometric series approach
• Used trig series of the form

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The Solution

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

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The Solution

• Began with

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The Solution

• Next

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The Solution

• Then a linear combination of the original
  solutions
• Constant

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The Solution

• Let be the root of the eigenvalue equation

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The Solution

• Then Constant such that
• and
• Then we can expand as trig series

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The Solution

• Finally
• Poincaré wrote his final solution to the system
  of differential equations as

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And it Goes on

• Lemmas, theorems,corollaries invariant integrals,
  proofs
• Im starting feel like the jury who studied the
  original 198 pages
• The rest of Poincarés solution was an attempt to
  generalize the solution for the n body problem

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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

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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.