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“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 BarrierDistanceRivalry
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
PoincaréHe was unanimously
chosen by the juryHis paper consisted
of 158 pagesThe importance of his
work was obviousThe 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–
iP
im
ijji rPP 3,2,1i
The Solution
The equations of motion– Based on Newton’s law of gravitation
3
13
133
23
12
122
221
2
r
qqmk
r
qqmk
dt
qd iiiii
3
23
233
23
12
212
222
2
r
qqmk
r
qqmk
dt
qd iiiii
3
23
323
23
13
312
223
2
r
qqmk
r
qqmk
dt
qd iiiii
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
12 k
i j
2ij
ji
r
mm
12
21
31
13
23
32
r
mm
r
mm
r
mmV
The Solution
Equations in the Hamiltonian form:
dt
dqmp ijiij
Vm
pH
i
ij
ji
2
23
1,
ij
ij
p
H
dt
dq
ij
ij
q
H
dt
dp
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
02
23
12
2
dt
qdm
dt
qdm ij
ii
iji
The Solution
Integrate twice
and are constants of integration
jjijiii
iji
i
BtAqmdt
qdmdt
3
2
23
1
jA jB
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:
211
2
12 dt
qdq
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:
211
2
12 dt
qdq
212
2
22 dt
qdq
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:
211
2
12 dt
qdq
212
2
22 dt
qdq
213
2
32 dt
qdq
The Solution
and
221
2
11 dt
qdq
The Solution
and
221
2
11 dt
qdq
222
2
21 dt
qdq
The Solution
and
Then add the two together to get
221
2
11 dt
qdq
222
2
21 dt
qdq 2
232
31 dt
qdq
03
12
12
222
2
1
3
1
i
iii
ii
ii dt
qdqm
dt
qdqm
The Solution
Permute cyclically the variable and integrate to obtain
1
3
1
23
32 C
dt
dqq
dt
dqqm
i
ii
iii
2
3
1
31
13 C
dt
dqq
dt
dqqm
i
ii
iii
3
3
1
12
21 C
dt
dqq
dt
dqqm
i
ii
iii
The Solution
Consider
Then
3
1
ik
ijkj
ikij r
rq
ij
iji q
V
dt
qdm
2
2
The Solution
Multiply by and sum to get
integrate
dt
dqij
dt
dV
dt
qdp
ji
ijij
3
1,2
2
CVm
p
ji i
ij
3
1,
2
2
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
txtx nn ,...,11
h x tht ii
ix
integer
2
k
ktht ii
The Solution
We’ll focus on this the most concise of his mathematical solutions
Trigonometric series approach– Used trig series of the form
...sin...sin
...cos...cos
1
1
nxBxB
nxAxAAxf
n
no
The Solution
Tried to find a general solution for the system of linear differential equations
coefficients are periodic functions of with period
nniii xxdt
dx.11. ...
2n ki.t 2
The Solution
Began with
txtx nini .1.1 ,...,
ni ,...,1
The Solution
Next
2,...,2 .1.1 txtx nini
ni ,...,1
The Solution
Then a linear combination of the original solutions
Constant
txAtAtx knnikiki ...11.. ,...,2
A
The Solution
Let be the root of the eigenvalue equation
1S
0
...
............
...
...
.2.1.
.22.21.2
.12.11.1
SAAA
ASAA
AASA
nnnn
n
n
The Solution
Then Constant such that
and
Then we can expand as trig series
kB
tSt ii .11.1 2
n
kikki Bt
1..1
tS i.11i.1
The Solution
Finally…– Poincaré wrote his final solution to the
system of differential equations as
tex it
i .11
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.
