“The Most Celebrated of all Dynamical Problems” History and Details to the Restricted Three Body...

“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

qq

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.