NEWTONIAN THREE-BODY PROBLEM

PERIODIC TRAJECTORIES AND TOPOLOGICAL CLASSIFICATION

Veljko Dmitrašinović ,Milovan Šuvakov

Laboratory for gaseous electronics
Center for non-equilibrium processes
Institute of physics Belgrad

Seminar at Beihang University, Beijing, 8th May 2013.

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My collaborators Pavle and Milovan Šuvakov - The engines driving this project

Talk based on a recent paper:

OUTLINE

MOTIVATION: a personal message to students
PROBLEM DEFINITION: review of the two-body problem
KNOWN ORBITS: classical and more recent
CLASSIFICATION OF ORBITS: braids, shape sphere
HUNT FOR NEW SOLUTIONS
CONCLUSIONS, OUTLOOK, OPEN QUESTIONS

[www.phdcomics.com]

PERSONAL MOTIVATION

A MESSAGE TO STUDENTS:
WHY (NOT) WORK ON AGE-OLD PROBLEMS?
IMPORTANCE OF NEW METHODS:
SYNERGY OF ANALYTIC AND NUMERICAL WORK

MOTIVATION

Confining three-quark potentials: the Y- and $\Delta$ string

[V. Dmitrašinović, T. Sato, M. Šuvakov, Eur Phys J C 62 383 (2009)]
[V. Dmitrašinović, T. Sato, M. Šuvakov, Phys Rev D 80 054501 (2009)]
Classical mechanical variant of the same problem
[M. Šuvakov, V. Dmitrašinović Phys Rev E 83 056603 (2011)]

TWO-BODY PROBLEM

Configuration space dimension: 2 x 2 = 4
Phase space dimension: 2 x 4 = 8
Conservation laws:
1. energy: 1
2. CM motion: 2 + 2
3. angular momentum: 1
4. Laplace-Runge-Lenz vector - 1
Maximally super-integrable system (solutions?)

THREE-BODY PROBLEM

Configuration space dimension: 3 x 2 = 6
Phase space dimension: 2 x 4 = 12
Conservation laws:
1. energy: 1
2. CM motion: 2 + 2
3. angular momentum: 1
4. remnant: 6 = 3 + 3
No more [H. Bruns, Acta Math. 11, 25 (1887) ]

THREE-BODY PROBLEM

Solutions can be:
- bounded / unbounded
- periodic / quasi-periodic / chaotic
H. Poincare:

... what makes these (periodic) solutions so precious to us, is that they are, so to say, the only opening through which we can try to penetrate in a place which, up to now, was supposed to be inaccessible

KNOWN ORBITS

Three classical orbits:
1. Free fall
2. Euler sln (1767), one body in the middle - always unstable
3. Lagrange sln (1772), stable only for very unequal masses
Only 2. and 3. collisionless
Only 3. (potentially) stable

Classical orbits in astronomy:

Lagrange orbits:
- Trojans: Achilles, Patroclus, Hector, Nestor (1905 and after)
- Sun - Mars - 5261 Eureka (1990), to date two more:
- SOHO and two missions to L1, and two to L2
Horseshoe orbit (in the restricted three-body problem) between two Lagrange points: L4 and L5
- Sun - Earth - 3753 Cruithne (1986)

MORE RECENT ORBITS

Space travel and modern electronic computers revived interest in this field in the 1960’s.
Two (infinite) families (angular mom. is the free parameter) of periodic three-body orbits were found in the mid-1970’s by Broucke and by Henon.
Interest subsided until 1990’s.

MOORE'S ORBITS

Moore found numerically several new orbits, among them:
[C. Moore, Phys Rev Lett 70 3675 (1993)]
A new orbit that belongs to the same family as Euler's and Lagrange slns and presents an equal mass analogon of the horseshoe orbit.
The famous "figure-8" solution

FIGURE-8

Chenciner and Montgomery rediscovered and formally proved existence in 2000.
[A. Chenciner, R. Montgomery, Ann. of Math. 152 881-901 (2000)]
We found this orbit in two other potentials ($\Delta$ and Y-string)
[M. Šuvakov, V. Dmitrašinović, Phys Rev E 83 056603 (2011)]
Moore's PRL was the first attempt at classification ...

MOORE'S FIGURE-8

APPLET

BRAIDS

JACOBI COORDINATES

$$\rho = \frac{1}{\sqrt{2}} (\mathbf{x}_1-\mathbf{x}_2)$$ $$\lambda = \frac{1}{\sqrt{6}} (\mathbf{x}_1+\mathbf{x}_2-2 \mathbf{x}_3)$$ Complicated representation of permutation symmetry!

SHAPE SPHERE

$$ \vec n = (n_x^{'},n_y^{'},n_z^{'}) = \left(\frac{2 {\vec \rho} \cdot {\vec \lambda}}{R^2}, \frac{\lambda^2 - \rho^2}{R^2}, \frac{2 ({\vec \rho} \times {\vec \lambda}) \cdot \vec e_z}{R^2} \right ) $$ $$R = \sqrt{\rho^{2} + \lambda^{2}}$$ Simple representation of permutation group!

CLASSIFICATION OF ORBITS

Sphere without three points ~ plane w/o two
Conjugation classes of a free group on two elements.
For better legibility $A=a^{-1}$, $B=b^{-1}$.
Broucke-Henon --- $a$
Moore's figure eight --- $abAB$

BROUCKE-HENON & EIGHT

HUNT FOR NEW SOLUTIONS: ASSUMPTIONS

Motion in a plane
Equal masses
Newtonian interaction ($-1/r$ potential)
Under these conditions: permutation symmetry important, angular momentum is a "scalar", and the two-body collision configurations are the only singularities of the potential.

METHODS

Runge-Kutta-Fehlberg method (RKF45)
Accumulated error during one period $\sim 10^{-12}$
Finding a solution is an optimisation problem
Phase space: 6-dimensional
- Scaling: -1
- Collinear configuration: -1
- Middle point: -1
- $R'(0)=0$: -1
Zero angular momentum $L=0$

SOLUTIONS

explain the classes!

BUTTERFLY I

MOTH I

DRAGONFLY

GOGGLES

YIN-YANG I

BUTTERFLY IV

HUNTING GROUNDS

CONCLUSIONS, OUTLOOK

We have:
- Found new solutions
- Used the topological classification method (first time)
Future:
- Check of stability
- Extend to different masses and angular momenta
- Other systems (He atom with Coulomb interaction)
Deeper questions: Why?