The N-Body Problem (NBD)

The Three-Body Problem

from a variational point of view

Kuo‐Chang Chen

kchen@math.nthu.edu.tw

Celestial Mechanics –

The N-Body Problem (NBD)

Ultimate Goal:

Understand the evolution of celestial bodies under the influence of

gravitation.

Kepler (1609,1619)

Three Laws of Planetary Motions

♦ The orbit of Mars is an ellipse with the Sun in one of its foci

♦ The line joining the planet to the Sun swept out equal areas in

equal times

♦ For any two planets the ratio of the squares of their periods is the same as the ratio of the cubes of the mean radii of their orbits

Everything happens as if matter attracts matter in direct proportion to the products of masses and in inverse proportion to the square of the distances.

Newton (Principia 1687)

Law of Universal Gravitation

Let be the position of mass . Equations of motions for the n-body problem:

Newton (Principia 1687)

Law of Universal Gravitation

Kepler Problem versus the N-Body Problem

n=2: Newton’s Law Kepler’s Laws ⇒

n 3 ? ≥

The three-body problem is of such importance in astronomy, and is at the same time so

difficult, that all efforts of geometers have long been directed toward it.

Poincaré (1892):

Major Questions on the N-Body Problem

♦

What kind of motions are possible and why?

Does Newton’s law along explains all

astronomical phenomena when relativistic effects are negligible?

♦

According to Newton’s law:

Is the solar system stable?

Is the Sun-Earth-Moon system stable?

♦

What kind of space mission designs can be

realized, given the constraint of current technology?

Will we ever be able to visit our neighboring star systems?

Let

be the potential energy.

Newton’s equations can be written

Hamiltonian Formulation

Let

be the kinetic energy,

be the Hamiltonian, where

Then Newton’s equations can be written

Let be the Lagrangian.

Then Newton’s equations are Euler-Lagrange equations of the action functional:

Lagrangian Formulation

Critical points of in are solutions of the n-body problem

(w/ or w/o collisions).

Why is the Three-Body Problem Difficult?

♦

The dimension of the system is high.

In the spatial (3BD), after reduction by

integrals of motions, it is a dynamical system on an 8-dimensional integral manifold.

♦

The system of equations is singular.

When n 3, generally collision singularities can not be regularized.

≥

♦

The action functional does not discriminate collision solutions from classical solutions.

Classical Results

♦

Euler (1767), Lagrange (1772):

Given any 3 masses, there are 5 classes of self-similar (homographic) solutions.

L1 ~ L5 are called Lagrange points or libration points

♦

Hill, Poincaré, Moulton, Birkhoff, Wintner, Hopf, Conley, … :

Existence of prograde, retrograde, and nearly circular solutions.

Either one mass is infinitesimal or one binary is tight.

♦

Sundman (1913), McGehee (1974):

Dynamics near triple collisions.

♦

Chazy (1922), Sitnikov (1959):

Complete classification of asymptotic behavior.

♦

Smale (1970), Meyer-McCord-Wang (1998):

Birkhoff conjecture for the (3BD).

Action of Collision Orbits

♦

Gordon (Am. J. Math. 1977):

Keplerian orbits with the same masses and least periods have the same action values.

♦

Poincaré (C.R.Acad.Sci. 1896):

Minimize among planar loops in a homology class. Collision loops can have finite action.

♦

Venturelli (C.R.Acad.Sci. 2001):

Action minimizers for (3BD) with homology constraints are either Lagrangian solutions, collision solutions, or do not exist.

Recent Progress – Figure-8 Orbit

♦

Chenciner-Montgomery (Ann.Math.2000):

Existence of Figure-8 orbit for the (3BD) with equal masses.

Idea of proof:

Minimize among loops in satisfying

Careful estimates show that

Principle of symmetric criticality (Palais 1979):

Minimizers in solve (3BD) provided all masses are equal.

Recent Progress – N-Body Problem

Chenciner-Venturelli (Cel.Mech.Dyn.Ast.2000) K.C. (Arch.Rat.Mech.Ana.2001)

K.C. (Erg.Thy.Dyn.Sys.2003) K.C. (Arch.Rat.Mech.Ana.2003)

Ferrario-Terracini (Invent.Math.2004)

Venturelli-Terracini (Arch.Rat.Mech.Ana.2007) Barutello-Terracini (Nonlinearity 2005)

Ferrario (Arch.Rat.Mech.Ana.2006) …..

Existence of miscellaneous solutions with some equal masses.

Marchal, Chenciner (Proc.I.C.M. 2002)

Action minimizers for problems with fixed ends are free from interior collisions.

Ferrario-Terracini (Invent.Math.2004)

Action minimizers for some equivariant problems are free from collisions.

K.C. (Arch.Rat.Mech.Ana.2006)

Action minimizers for problems with free boundaries are free from collisions.

Question:

Are variational methods useful only when some masses are equal?

For “most” choices of masses, there exist  infinitely many periodic and quasi‐periodic retrograde solutions for the (3BD), none of which contain a tight binary.

Retrograde Orbits with Various Masses

♦

K.C. (Annals Math., 2008):

The (3BD) has infinitely many periodic and quasi-periodic retrograde solutions without tight binaries

provided , where

.

Retrograde Orbits with Various Masses

♦

K.C. (Annals Math., 2008):

Region of admissible masses

Key idea of proof:

Let , . On

the action functional can be written

Then

Can show that this lower bound estimate is quite sharp and is (a little bit) higher than Kepler-like retrograde paths as long as ,

which is valid for most choices of masses.

THE END