“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 q2 i
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
 2
rij
 Potential energy of the entire system
m2 m3 m3 m1 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 2 qij
i
i 1

dt
A j and B j
integration
2
3
  mi qij  A j 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
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:
d 2 q11
d 2 q13
d 2 q12
 q12
 q32
 q22
2
2
dt
dt 2
dt
The Solution
 and
d 2 q21
q11
dt 2
The Solution
 and
d 2 q21
q11
dt 2
d 2 q22
q21
dt 2
The Solution
 and
d 2 q21
q11
dt 2
d 2 q22
q21
dt 2
d 2 q23
q31
dt 2
 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
dt 2
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 xi 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 cos nx  ...
 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
A1.2
...
A1.n
A2.2  S ...
A2.n
...
...
...
...
An.1
An.2
... An.n  S
0
The Solution
 Then Constant Bk such that
1.i t  2   S11.i t 
n
and 1.i t    Bk k .i
k 1
 Then we can expand S11.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.